Rare numbers
You are encouraged to solve this task according to the task description, using any language you may know.
- Definitions and restrictions
Rare numbers are positive integers n where:
- n is expressed in base ten
- r is the reverse of n (decimal digits)
- n must be non-palindromic (n ≠ r)
- (n+r) is the sum
- (n-r) is the difference and must be positive
- the sum and the difference must be perfect squares
- Task
-
- find and show the first 5 rare numbers
- find and show the first 8 rare numbers (optional)
- find and show more rare numbers (stretch goal)
Show all output here, on this page.
- References
-
- an OEIS entry: A035519 rare numbers.
- an OEIS entry: A059755 odd rare numbers.
- planetmath entry: rare numbers. (some hints)
- author's website: rare numbers by Shyam Sunder Gupta. (lots of hints and some observations).
C#
Traditional
Converted to unsigned longs in order to reach 19 digits. <lang csharp>using System; using System.Collections.Generic; using System.Linq; using static System.Console; using UI = System.UInt64; using LST = System.Collections.Generic.List<System.Collections.Generic.List<sbyte>>; using Lst = System.Collections.Generic.List<sbyte>; using DT = System.DateTime;
class Program {
const sbyte MxD = 19;
public struct term { public UI coeff; public sbyte a, b;
public term(UI c, int a_, int b_) { coeff = c; a = (sbyte)a_; b = (sbyte)b_; } }
static int[] digs; static List<UI> res; static sbyte count = 0; static DT st; static List<List<term>> tLst; static List<LST> lists; static Dictionary<int, LST> fml, dmd; static Lst dl, zl, el, ol, il; static bool odd; static int nd, nd2; static LST ixs; static int[] cnd, di; static LST dis; static UI Dif;
// converts digs array to the "difference"
static UI ToDif() { UI r = 0; for (int i = 0; i < digs.Length; i++)
r = r * 10 + (uint)digs[i]; return r; }
// converts digs array to the "sum"
static UI ToSum() { UI r = 0; for (int i = digs.Length - 1; i >= 0; i--)
r = r * 10 + (uint)digs[i]; return Dif + (r << 1); }
// determines if the nmbr is square or not
static bool IsSquare(UI nmbr) { if ((0x202021202030213 & (1 << (int)(nmbr & 63))) != 0)
{ UI r = (UI)Math.Sqrt((double)nmbr); return r * r == nmbr; } return false; }
// returns sequence of sbytes
static Lst Seq(sbyte from, int to, sbyte stp = 1) { Lst res = new Lst();
for (sbyte item = from; item <= to; item += stp) res.Add(item); return res; }
// Recursive closure to generate (n+r) candidates from (n-r) candidates
static void Fnpr(int lev) { if (lev == dis.Count) { digs[ixs[0][0]] = fml[cnd[0]][di[0]][0];
digs[ixs[0][1]] = fml[cnd[0]][di[0]][1]; int le = di.Length, i = 1;
if (odd) digs[nd >> 1] = di[--le]; foreach (sbyte d in di.Skip(1).Take(le - 1)) {
digs[ixs[i][0]] = dmd[cnd[i]][d][0]; digs[ixs[i][1]] = dmd[cnd[i++]][d][1]; }
if (!IsSquare(ToSum())) return; res.Add(ToDif()); WriteLine("{0,16:n0}{1,4} ({2:n0})",
(DT.Now - st).TotalMilliseconds, ++count, res.Last()); }
else foreach (var n in dis[lev]) { di[lev] = n; Fnpr(lev + 1); } }
// Recursive closure to generate (n-r) candidates with a given number of digits.
static void Fnmr (LST list, int lev) { if (lev == list.Count) { Dif = 0; sbyte i = 0;
foreach (var t in tLst[nd2]) { if (cnd[i] < 0) Dif -= t.coeff * (UI)(-cnd[i++]);
else Dif += t.coeff * (UI)cnd[i++]; } if (Dif <= 0 || !IsSquare(Dif)) return;
dis = new LST { Seq(0, fml[cnd[0]].Count - 1) };
foreach (int ii in cnd.Skip(1)) dis.Add(Seq(0, dmd[ii].Count - 1));
if (odd) dis.Add(il); di = new int[dis.Count]; Fnpr(0);
} else foreach(sbyte n in list[lev]) { cnd[lev] = n; Fnmr(list, lev + 1); } }
static void init() { UI pow = 1;
// terms of (n-r) expression for number of digits from 2 to maxDigits
tLst = new List<List<term>>(); foreach (int r in Seq(2, MxD)) {
List<term> terms = new List<term>(); pow *= 10; UI p1 = pow, p2 = 1;
for (int i1 = 0, i2 = r - 1; i1 < i2; i1++, i2--) {
terms.Add(new term(p1 - p2, i1, i2)); p1 /= 10; p2 *= 10; }
tLst.Add(terms); }
// map of first minus last digits for 'n' to pairs giving this value
fml = new Dictionary<int, LST> {
[0] = new LST { new Lst { 2, 2 }, new Lst { 8, 8 } },
[1] = new LST { new Lst { 6, 5 }, new Lst { 8, 7 } },
[4] = new LST { new Lst { 4, 0 } },
[6] = new LST { new Lst { 6, 0 }, new Lst { 8, 2 } } };
// map of other digit differences for 'n' to pairs giving this value
dmd = new Dictionary<int, LST>();
for (sbyte i = 0; i < 10; i++) for (sbyte j = 0, d = i; j < 10; j++, d--) {
if (dmd.ContainsKey(d)) dmd[d].Add(new Lst { i, j });
else dmd[d] = new LST { new Lst { i, j } }; }
dl = Seq(-9, 9); // all differences
zl = Seq( 0, 0); // zero differences only
el = Seq(-8, 8, 2); // even differences only
ol = Seq(-9, 9, 2); // odd differences only
il = Seq( 0, 9); lists = new List<LST>();
foreach (sbyte f in fml.Keys) lists.Add(new LST { new Lst { f } }); }
static void Main(string[] args) { init(); res = new List<UI>(); st = DT.Now; count = 0;
WriteLine("{0,5}{1,12}{2,4}{3,14}", "digs", "elapsed(ms)", "R/N", "Unordered Rare Numbers");
for (nd = 2, nd2 = 0, odd = false; nd <= MxD; nd++, nd2++, odd = !odd) { digs = new int[nd];
if (nd == 4) { lists[0].Add(zl); lists[1].Add(ol); lists[2].Add(el); lists[3].Add(ol); }
else if (tLst[nd2].Count > lists[0].Count) foreach (LST list in lists) list.Add(dl);
ixs = new LST();
foreach (term t in tLst[nd2]) ixs.Add(new Lst { t.a, t.b });
foreach (LST list in lists) { cnd = new int[list.Count]; Fnmr(list, 0); }
WriteLine(" {0,2} {1,10:n0}", nd, (DT.Now - st).TotalMilliseconds); }
res.Sort();
WriteLine("\nThe {0} rare numbers with up to {1} digits are:", res.Count, MxD);
count = 0; foreach (var rare in res) WriteLine("{0,2}:{1,27:n0}", ++count, rare);
if (System.Diagnostics.Debugger.IsAttached) ReadKey(); }
}</lang>
- Output:
Results from a core i7-7700 @ 3.6Ghz. This C# version isn't as fast as the Go version using the same hardware. C# computes up to 17, 18 and 19 digits in under 9 minutes, 1 2/3 hours and over 2 1/2 hours respectively. (Go is about 6 minutes, 1 1/4 hours, and under 2 hours).
The long-to-ulong conversion isn't causing the reduced performance, C# has more overhead as compared to Go. This C# version can easily be converted to use BigIntegers to go beyond 19 digits, but becomes around eight times slower. (ugh!)
digs elapsed(ms) R/N Rare Numbers
27 1 (65)
2 28
3 28
4 29
5 29
29 2 (621,770)
6 29
7 30
8 34
34 3 (281,089,082)
9 36
36 4 (2,022,652,202)
61 5 (2,042,832,002)
10 121
11 176
448 6 (872,546,974,178)
481 7 (872,568,754,178)
935 8 (868,591,084,757)
12 1,232
1,577 9 (6,979,302,951,885)
13 2,087
6,274 10 (20,313,693,904,202)
6,351 11 (20,313,839,704,202)
8,039 12 (20,331,657,922,202)
8,292 13 (20,331,875,722,202)
9,000 14 (20,333,875,702,202)
21,212 15 (40,313,893,704,200)
21,365 16 (40,351,893,720,200)
14 23,898
23,964 17 (200,142,385,731,002)
24,198 18 (221,462,345,754,122)
27,241 19 (816,984,566,129,618)
28,834 20 (245,518,996,076,442)
29,074 21 (204,238,494,066,002)
29,147 22 (248,359,494,187,442)
29,476 23 (244,062,891,224,042)
35,481 24 (403,058,392,434,500)
35,721 25 (441,054,594,034,340)
15 38,231
92,116 26 (2,133,786,945,766,212)
113,469 27 (2,135,568,943,984,212)
116,787 28 (8,191,154,686,620,818)
119,647 29 (8,191,156,864,620,818)
120,912 30 (2,135,764,587,964,212)
122,735 31 (2,135,786,765,764,212)
127,126 32 (8,191,376,864,400,818)
141,793 33 (2,078,311,262,161,202)
179,832 34 (8,052,956,026,592,517)
184,647 35 (8,052,956,206,592,517)
221,279 36 (8,650,327,689,541,457)
223,721 37 (8,650,349,867,341,457)
225,520 38 (6,157,577,986,646,405)
273,238 39 (4,135,786,945,764,210)
312,969 40 (6,889,765,708,183,410)
16 316,349
322,961 41 (86,965,750,494,756,968)
323,958 42 (22,542,040,692,914,522)
502,805 43 (67,725,910,561,765,640)
17 519,583
576,058 44 (284,684,666,566,486,482)
707,530 45 (225,342,456,863,243,522)
756,188 46 (225,342,458,663,243,522)
856,346 47 (225,342,478,643,243,522)
928,546 48 (284,684,868,364,486,482)
1,311,170 49 (871,975,098,681,469,178)
2,031,664 50 (865,721,270,017,296,468)
2,048,209 51 (297,128,548,234,950,692)
2,057,281 52 (297,128,722,852,950,692)
2,164,878 53 (811,865,096,390,477,018)
2,217,508 54 (297,148,324,656,930,692)
2,242,999 55 (297,148,546,434,930,692)
2,576,805 56 (898,907,259,301,737,498)
3,169,675 57 (631,688,638,047,992,345)
3,200,223 58 (619,431,353,040,136,925)
3,482,517 59 (619,631,153,042,134,925)
3,550,566 60 (633,288,858,025,996,145)
3,623,653 61 (633,488,632,647,994,145)
4,605,503 62 (653,488,856,225,994,125)
5,198,241 63 (497,168,548,234,910,690)
18 6,028,721
6,130,826 64 (2,551,755,006,254,571,552)
6,152,283 65 (2,702,373,360,882,732,072)
6,424,945 66 (2,825,378,427,312,735,282)
6,447,566 67 (8,066,308,349,502,036,608)
6,677,925 68 (2,042,401,829,204,402,402)
6,725,119 69 (2,420,424,089,100,600,242)
6,843,016 70 (8,320,411,466,598,809,138)
7,161,527 71 (8,197,906,905,009,010,818)
7,198,112 72 (2,060,303,819,041,450,202)
7,450,028 73 (8,200,756,128,308,135,597)
7,881,502 74 (6,531,727,101,458,000,045)
9,234,318 75 (6,988,066,446,726,832,640)
19 9,394,513
The 75 rare numbers with up to 19 digits are:
1: 65
2: 621,770
3: 281,089,082
4: 2,022,652,202
5: 2,042,832,002
6: 868,591,084,757
7: 872,546,974,178
8: 872,568,754,178
9: 6,979,302,951,885
10: 20,313,693,904,202
11: 20,313,839,704,202
12: 20,331,657,922,202
13: 20,331,875,722,202
14: 20,333,875,702,202
15: 40,313,893,704,200
16: 40,351,893,720,200
17: 200,142,385,731,002
18: 204,238,494,066,002
19: 221,462,345,754,122
20: 244,062,891,224,042
21: 245,518,996,076,442
22: 248,359,494,187,442
23: 403,058,392,434,500
24: 441,054,594,034,340
25: 816,984,566,129,618
26: 2,078,311,262,161,202
27: 2,133,786,945,766,212
28: 2,135,568,943,984,212
29: 2,135,764,587,964,212
30: 2,135,786,765,764,212
31: 4,135,786,945,764,210
32: 6,157,577,986,646,405
33: 6,889,765,708,183,410
34: 8,052,956,026,592,517
35: 8,052,956,206,592,517
36: 8,191,154,686,620,818
37: 8,191,156,864,620,818
38: 8,191,376,864,400,818
39: 8,650,327,689,541,457
40: 8,650,349,867,341,457
41: 22,542,040,692,914,522
42: 67,725,910,561,765,640
43: 86,965,750,494,756,968
44: 225,342,456,863,243,522
45: 225,342,458,663,243,522
46: 225,342,478,643,243,522
47: 284,684,666,566,486,482
48: 284,684,868,364,486,482
49: 297,128,548,234,950,692
50: 297,128,722,852,950,692
51: 297,148,324,656,930,692
52: 297,148,546,434,930,692
53: 497,168,548,234,910,690
54: 619,431,353,040,136,925
55: 619,631,153,042,134,925
56: 631,688,638,047,992,345
57: 633,288,858,025,996,145
58: 633,488,632,647,994,145
59: 653,488,856,225,994,125
60: 811,865,096,390,477,018
61: 865,721,270,017,296,468
62: 871,975,098,681,469,178
63: 898,907,259,301,737,498
64: 2,042,401,829,204,402,402
65: 2,060,303,819,041,450,202
66: 2,420,424,089,100,600,242
67: 2,551,755,006,254,571,552
68: 2,702,373,360,882,732,072
69: 2,825,378,427,312,735,282
70: 6,531,727,101,458,000,045
71: 6,988,066,446,726,832,640
72: 8,066,308,349,502,036,608
73: 8,197,906,905,009,010,818
74: 8,200,756,128,308,135,597
75: 8,320,411,466,598,809,138
Quicker
Along the lines of the C++ version. Computing the possible squares for the sums and differences, then comparing them together and reporting the ones that have a proper forward, reverse result. To save computation time, some shortcuts have been taken to avoid generating many non-square numbers.
Update, added computation of digital root, which increased performance a few percentage points. Interestingly, the digital root is always zero for the difference list of squares, so no advantage is given by tracking it. Only the sum list of squares benefits from calculating the digital root of the partial sum and using an abbreviated sequence for the last round of permutation.
<lang csharp>using static System.Math; // for Sqrt()
using System.Collections.Generic; // for List<>, .Count
using System.Linq; // for .Last(), .ToList()
using System.Diagnostics; // for Stopwatch()
using static System.Console; // for Write(), WriteLine()
using llst = System.Collections.Generic.List<int[]>;
class Program
{
#region vars
static int[] d, // permutation working array
drar = new int[19], // digital root lookup array
dac; // running digital root array
static long[] p = new long[20], // powers of 10
ac, // accumulator array
pp; // long coefficient array that combines with digits of working array
static bool odd = false; // flag for odd number of digits
static long sum, // calculated sum of terms (square candidate)
rt; // root of sum
static int cn = 0, // solution counter
nd = 2, // number of digits
nd1 = nd - 1, // nd helper
ln, // previous value of "n" (in Recurse())
dl; // length of "d" array;
static Stopwatch sw = new Stopwatch(), swt = new Stopwatch(); // for timings
static List<long> sr = new List<long>(); // temporary list of squares used for building
static readonly int[] tlo = new int[] { 0, 1, 4, 5, 6 }, // primary differences starting point
all = Seq(-9, 9), // all possible differences
odl = Seq(-9, 9, 2), // odd possible differences
evl = Seq(-8, 8, 2), // even possible differences
thi = new int[] { 4, 5, 6, 9, 10, 11, 14, 15, 16 }, // primary sums staring point. note: (0, 1) omitted, as any square generated will not have enough digits
alh = Seq(0, 18), // all possible sums
odh = Seq(1, 17, 2), // odd possible sums
evh = Seq(0, 18, 2), // even possible sums
ten = Seq(0, 9), // used for odd number of digits
z = Seq(0, 0), // no difference, used to avoid generating a bunch of negative square candidates
t7 = new int[] { -3, 7 }, // shortcut for low 5
nin = new int[] { 9 }, // shortcut for hi 10
tn = new int[] { 10 }, // shortcut for hi 0 (unused, uneeded)
t12 = new int[] { 2, 12 }, // shortcut for hi 5
o11 = new int[] { 1, 11 }, // shortcut for hi 15
pos = new int[] { 0, 1, 4, 5, 6, 9 }; // shortcut for 2nd lo 0
static llst lul = new llst { z, odl, null, null, evl, t7, odl }, // shortcut lookup lo primary
luh = new llst { tn, evh, null, null, evh, t12, odh, null, null, evh, nin, odh, null, null, odh, o11, evh }, // shortcut lookup hi primary
l2l = new llst { pos, null, null, null, all, null, all }, // shortcut lookup lo secondary
l2h = new llst { null, null, null, null, alh, null, alh, null, null, null, alh, null, null, null, alh, null, alh }, lu, l2; // shortcut lookup hi secondary
static int[][] chTen = new int[][] { new int[] { 0,2,5,8,9 }, new int[] { 0,3,4,6,9 }, new int[] { 1,4,7,8 }, new int[] { 2,3,5,8 },
new int[] { 0,3,6,7,9 }, new int[] { 1,2,4,7 }, new int[] { 2,5,6,8 }, new int[] { 0,1,3,6,9 }, new int[] { 1,4,5,7 } };
static int[][] chAH = new int[][] { new int[] { 0,2,5,8,9,11,14,17,18 }, new int[] { 0,3,4,6,9,12,13,15,18 }, new int[] { 1,4,7,8,10,13,16,17 },
new int[] { 2,3,5,8,11,12,14,17 }, new int[] { 0,3,6,7,9,12,15,16,18 }, new int[] { 1,2,4,7,10,11,13,16 },
new int[] { 2,5,6,8,11,14,15,17 }, new int[] { 0,1,3,6,9,10,12,15,18 }, new int[] { 1,4,5,7,10,13,14,16 } };
#endregion vars
// Returns a sequence of integers
static int[] Seq(int f, int t, int s = 1) { int[] r = new int[(t - f) / s + 1]; for (int i = 0; i < r.Length; i++, f += s) r[i] = f; return r; }
// Returns Integer Square Root
static long ISR(long s) { return (long)Sqrt(s); }
// Recursively determines whether "r" is the reverse of "f"
static bool IsRev(int nd, long f, long r) { nd--; return f / p[nd] != r % 10 ? false : (nd < 1 ? true : IsRev(nd, f % p[nd], r / 10)); }
// Recursive procedure to evaluate the permutations, no shortcuts
static void RecurseLE5(llst lst, int lv) { if (lv == dl) { // check if on last stage of permutation
if ((sum = ac[lv - 1]) > 0) if ((rt = (long)Sqrt(sum)) * rt == sum) sr.Add(sum); } // test accumulated sum, append to result if square
else foreach (int n in lst[lv]) { // set up next permutation
d[lv] = n; if (lv == 0) ac[0] = pp[0] * n; else ac[lv] = ac[lv - 1] + pp[lv] * n; // update accumulated sum
RecurseLE5(lst, lv + 1); } } // Recursively call next level
// Recursive procedure to evaluate the hi permutations, shortcuts added to avoid generating many non-squares, digital root calc added
static void Recursehi(llst lst, int lv) {
int lv1 = lv - 1; if (lv == dl) { // check if on last stage of permutation
if ((0x202021202030213 & (1 << (int)((sum = ac[lv1]) & 63))) != 0) // test accumulated sum, append to result if square
if ((rt = (long)Sqrt(sum)) * rt == sum) sr.Add(sum); }
else foreach (int n in lst[lv]) { // set up next permutation
d[lv] = n; if (lv == 0) { ac[0] = pp[0] * n; dac[0] = drar[n]; } // update accumulated sum and running dr
else { ac[lv] = ac[lv1] + pp[lv] * n; dac[lv] = dac[lv1] + drar[n]; if (dac[lv] > 8) dac[lv] -= 9; }
switch (lv) { // shortcuts to be performed on designated levels
case 0: lst[1] = lu[ln = n]; lst[2] = l2[n]; break; // primary level: set shortcuts for secondary level
case 1: // secondary level: set shortcuts for tertiary level
switch (ln) { // for sums
case 5: case 15: lst[2] = n < 10 ? evh : odh; break;
case 9: lst[2] = ((n >> 1) & 1) == 0 ? evh : odh; break;
case 11: lst[2] = ((n >> 1) & 1) == 1 ? evh : odh; break; } break; }
if (lv == dl - 2) lst[dl - 1] = odd ? chTen[dac[dl - 2]] : chAH[dac[dl - 2]]; // reduce last round according to dr calc
Recursehi(lst, lv + 1); } } // Recursively call next level
// Recursive procedure to evaluate the lo permutations, shortcuts added to avoid generating many non-squares
static void Recurselo(llst lst, int lv) { int lv1 = lv - 1; if (lv == dl) { // check if on last stage of permutation
if ((sum = ac[lv1]) > 0) if ((rt = (long)Sqrt(sum)) * rt == sum) sr.Add(sum); } // test accumulated sum, append to result if square
else foreach (int n in lst[lv]) { // set up next permutation
d[lv] = n; if (lv == 0) ac[0] = pp[0] * n; else ac[lv] = ac[lv1] + pp[lv] * n; // update accumulated sum
switch (lv) { // shortcuts to be performed on designated levels
case 0: lst[1] = lu[ln = n]; lst[2] = l2[n]; break; // primary level: set shortcuts for secondary level
case 1: // secondary level: set shortcuts for tertiary level
switch (ln) { // for difs
case 1: lst[2] = (((n + 9) >> 1) & 1) == 0 ? evl : odl; break;
case 5: lst[2] = n < 0 ? evl : odl; break; } break; }
Recurselo(lst, lv + 1); } } // Recursively call next level
// Produces a list of candidate square numbers
static List<long> listEm(llst lst, llst plu, llst pl2) {
d = new int[dl = lst.Count]; sr.Clear(); lu = plu; l2 = pl2; ac = new long[dl]; dac = new int[dl]; // init support vars
pp = new long[dl]; for (int i = 0, j = nd1; i < dl; i++, j--) pp[i] = lst[0].Length > 6 ? p[j] + p[i] : p[j] - p[i]; // build coefficients array
if (nd <= 5) RecurseLE5(lst, 0); else { if (lst[0].Length > 8) Recursehi(lst, 0); else Recurselo(lst, 0); } return sr; } // call appropriate recursive procedure
// Reveals whether combining two lists of squares can produce a Rare number
static void Reveal(List<long> lo, List<long> hi) { List<string> s = new List<string>(); // create temp list of results
foreach (long l in lo) foreach (long h in hi) { long r = (h - l) >> 1, f = h - r; // generate all possible fwd & rev candidates from lists
if (IsRev(nd, f, r)) s.Add(string.Format("{0,20} {1,11} {2,10} ", f, ISR(h), ISR(l))); } // test and append sucesses to temp list
s.Sort(); if (s.Count > 0) foreach (string t in s) // if there are any, output sorted results
Write("{0,2} {1}{2}", ++cn, t, t == s.Last() ? "" : "\n"); else Write("{0,48}", ""); }
static void Main(string[] args) {
WriteLine("{0,3}{1,20} {2,11} {3,10} {4,4}{5,16} {6, 17}", "nth", "forward", "rt.sum", "rt.dif", "digs", "block time", "total time");
p[0] = 1; for (int i = 0, j = 1; j < p.Length; i = j++) p[j] = p[i] * 10; // create powers of 10 array
for (int i = 0; i < drar.Length; i++) drar[i] = (i << 1) % 9; // create digital root array
llst lls = new llst { tlo }, hls = new llst { thi }; sw.Start(); swt.Start(); // initialize permutations list, timers
for (; nd <= 18; nd1 = nd++, odd = !odd) { // loop through all numbers of digits
if (nd > 2) if (odd) hls.Add(ten); else { lls.Add(all); hls[hls.Count - 1] = alh; } // build permutations list
Reveal(listEm(lls, lul, l2l).ToList(), listEm(hls, luh, l2h)); // reveal results
if (!odd && nd > 5) hls[hls.Count - 1] = alh; // restore last element of hls, so that dr shortcut doesn't mess up next nd
WriteLine("{0,2}: {1} {2}", nd, sw.Elapsed, swt.Elapsed); sw.Restart(); }
// 19
hls.Add(ten);
Reveal(listEmU(lls, lul, l2l).ToList(), listEmU(hls, luh, l2h)); // reveal unsigned results
WriteLine("{0,2}: {1} {2}", nd, sw.Elapsed, swt.Elapsed);
}
#region 19
static ulong usum, // unsigned calculated sum of terms (square candidate)
urt; // unsigned root of sum
static ulong[] acu, // unsigned accumulator array
ppu; // unsigned long coefficient array that combines with digits of working array
static List<ulong> sru = new List<ulong>(); // unsigned temporary list of squares used for building
// Reveals whether combining two lists of unsigned squares can produce a Rare number
static void Reveal(List<ulong> lo, List<ulong> hi) {
List<string> s = new List<string>(); // create temp list of results
foreach (ulong l in lo) foreach (ulong h in hi) { ulong r = (h - l) >> 1, f = h - r; // generate all possible fwd & rev candidates from lists
if (IsRev(nd, f, r)) s.Add(string.Format("{0,20} {1,11} {2,10} ", f, ISR(h), ISR(l))); } // test and append sucesses to temp list
s.Sort(); if (s.Count > 0) foreach (string t in s) // if there are any, output sorted results
Write("{0,2} {1}{2}", ++cn, t, t == s.Last() ? "" : "\n"); else Write("{0,48}", ""); }
// Produces a list of unsigned candidate square numbers
static List<ulong> listEmU(llst lst, llst plu, llst pl2) {
d = new int[dl = lst.Count]; sru.Clear(); lu = plu; l2 = pl2; acu = new ulong[dl]; dac = new int[dl]; // init support vars
ppu = new ulong[dl]; for (int i = 0, j = nd1; i < dl; i++, j--) ppu[i] = (ulong)(lst[0].Length > 6 ? p[j] + p[i] : p[j] - p[i]); // build coefficients array
if (lst[0].Length > 8) RecurseUhi(lst, 0); else RecurseUlo(lst, 0); return sru; } // call recursive procedure
// Recursive procedure to evaluate the unsigned hi permutations, shortcuts added to avoid generating many non-squares, digital root calc added
static void RecurseUhi(llst lst, int lv) { int lv1 = lv - 1; if (lv == dl) { // check if on last stage of permutation
if ((0x202021202030213 & (1 << (int)((usum = acu[lv1]) & 63))) != 0) // test accumulated sum, append to result if square
if ((urt = (ulong)Sqrt(usum)) * urt == usum) sru.Add(usum); }
else foreach (int n in lst[lv]) { // set up next permutation
d[lv] = n; if (lv == 0) { acu[0] = ppu[0] * (uint)n; dac[0] = drar[n]; } // update accumulated sum and running dr
else { acu[lv] = n >= 0 ? acu[lv1] + ppu[lv] * (uint)n : acu[lv1] - ppu[lv] * (uint)-n; dac[lv] = dac[lv1] + drar[n]; if (dac[lv] > 8) dac[lv] -= 9; }
switch (lv) { // shortcuts to be performed on designated levels
case 0: lst[1] = lu[ln = n]; lst[2] = l2[n]; break; // primary level: set shortcuts for secondary level
case 1: // secondary level: set shortcuts for tertiary level
switch (ln) { // for sums
case 5: case 15: lst[2] = n < 10 ? evh : odh; break;
case 9: lst[2] = ((n >> 1) & 1) == 0 ? evh : odh; break;
case 11: lst[2] = ((n >> 1) & 1) == 1 ? evh : odh; break; } break; }
if (lv == dl - 2) lst[dl - 1] = odd ? chTen[dac[dl - 2]] : chAH[dac[dl - 2]]; // reduce last round according to dr calc
RecurseUhi(lst, lv + 1); } } // Recursively call next level
// Recursive procedure to evaluate the unsigned lo permutations, shortcuts added to avoid generating many non-squares
static void RecurseUlo(llst lst, int lv) { int lv1 = lv - 1; if (lv == dl) { // check if on last stage of permutation
if ((usum = acu[lv1]) > 0) if ((urt = (ulong)Sqrt(usum)) * urt == usum) sru.Add(usum); } // test accumulated sum, append to result if square
else foreach (int n in lst[lv]) { // set up next permutation
d[lv] = n; if (lv == 0) acu[0] = ppu[0] * (uint)n;
else acu[lv] = n >= 0 ? acu[lv1] + ppu[lv] * (uint)n : acu[lv1] - ppu[lv] * (uint)-n; // update accumulated sum
switch (lv) { // shortcuts to be performed on designated levels
case 0: lst[1] = lu[ln = n]; lst[2] = l2[n]; break; // primary level: set shortcuts for secondary level
case 1: // secondary level: set shortcuts for tertiary level
switch (ln) { // for difs
case 1: lst[2] = (((n + 9) >> 1) & 1) == 0 ? evl : odl; break;
case 5: lst[2] = n < 0 ? evl : odl; break; } break; }
RecurseUlo(lst, lv + 1); } } // Recursively call next level
// Returns unsigned Integer Square Root
static ulong ISR(ulong s) { return (ulong)Sqrt(s); }
// Recursively determines whether "r" is the reverse of "f"
static bool IsRev(int nd, ulong f, ulong r) { nd--; return f / (ulong)p[nd] != r % 10 ? false : (nd < 1 ? true : IsRev(nd, f % (ulong)p[nd], r / 10)); }
#endregion 19
}</lang>
- Output:
Results on the core i7-7700 @ 3.6Ghz.
nth forward rt.sum rt.dif digs block time total time
1 65 11 3 2: 00:00:00.0030626 00:00:00.0030626
3: 00:00:00.0001018 00:00:00.0033254
4: 00:00:00.0000963 00:00:00.0035054
5: 00:00:00.0000928 00:00:00.0036834
2 621770 836 738 6: 00:00:00.0021741 00:00:00.0059392
7: 00:00:00.0001724 00:00:00.0061956
8: 00:00:00.0002609 00:00:00.0065384
3 281089082 23708 330 9: 00:00:00.0012672 00:00:00.0079061
4 2022652202 63602 300
5 2042832002 63602 6360 10: 00:00:00.0045628 00:00:00.0125626
11: 00:00:00.0201361 00:00:00.0328037
6 868591084757 1275175 333333
7 872546974178 1320616 32670
8 872568754178 1320616 33330 12: 00:00:00.0519065 00:00:00.0848320
9 6979302951885 3586209 1047717 13: 00:00:00.3772503 00:00:00.4622089
10 20313693904202 6368252 269730
11 20313839704202 6368252 270270
12 20331657922202 6368252 329670
13 20331875722202 6368252 330330
14 20333875702202 6368252 336330
15 40313893704200 6368252 6330336
16 40351893720200 6368252 6336336 14: 00:00:00.9416903 00:00:01.4041338
17 200142385731002 20006998 69300
18 204238494066002 20122102 1891560
19 221462345754122 21045662 69300
20 244062891224042 22011022 1908060
21 245518996076442 22140228 921030
22 248359494187442 22206778 1891560
23 403058392434500 20211202 19940514
24 441054594034340 22011022 19940514
25 816984566129618 40421606 250800 15: 00:00:07.0248881 00:00:08.4296936
26 2078311262161202 64030648 7529850
27 2133786945766212 65272218 2666730
28 2135568943984212 65272218 3267330
29 2135764587964212 65272218 3326670
30 2135786765764212 65272218 3333330
31 4135786945764210 65272218 63333336
32 6157577986646405 105849161 33333333
33 6889765708183410 83866464 82133718
34 8052956026592517 123312255 29999997
35 8052956206592517 123312255 30000003
36 8191154686620818 127950856 3299670
37 8191156864620818 127950856 3300330
38 8191376864400818 127950856 3366330
39 8650327689541457 127246955 33299667
40 8650349867341457 127246955 33300333 16: 00:00:18.1046570 00:00:26.5344137
41 22542040692914522 212329862 333300
42 67725910561765640 269040196 251135808
43 86965750494756968 417050956 33000 17: 00:02:11.8544100 00:02:38.3889020
44 225342456863243522 671330638 297000
45 225342458663243522 671330638 303000
46 225342478643243522 671330638 363000
47 284684666566486482 754565658 30000
48 284684868364486482 754565658 636000
49 297128548234950692 770186978 32697330
50 297128722852950692 770186978 32702670
51 297148324656930692 770186978 33296670
52 297148546434930692 770186978 33303330
53 497168548234910690 770186978 633363336
54 619431353040136925 1071943279 299667003
55 619631153042134925 1071943279 300333003
56 631688638047992345 1083968809 297302703
57 633288858025996145 1083968809 302637303
58 633488632647994145 1083968809 303296697
59 653488856225994125 1083968809 363303363
60 811865096390477018 1273828556 33030330
61 865721270017296468 1315452006 32071170
62 871975098681469178 1320582934 3303300
63 898907259301737498 1339270086 64576740 18: 00:05:38.5737725 00:08:16.9627994
64 2042401829204402402 2021001202 18915600
65 2060303819041450202 2020110202 199405140
66 2420424089100600242 2200110022 19080600
67 2551755006254571552 2259094848 693000
68 2702373360882732072 2324811012 693000
69 2825378427312735282 2377130742 2508000
70 6531727101458000045 3454234451 1063822617
71 6988066446726832640 2729551744 2554541088
72 8066308349502036608 4016542096 2508000
73 8197906905009010818 4046976144 133408770
74 8200756128308135597 4019461925 495417087
75 8320411466598809138 4079154376 36366330 19: 00:42:31.7490390 00:50:48.7120790
C++
Calculate L and H independently
<lang cpp> // Rare Numbers : Nigel Galloway - December 20th., 2019; // Nigel Galloway/Enter your username - January 4th., 2021 (see discussion page.
- include <functional>
- include <bitset>
- include <cmath>
using namespace std; using Z2 = optional<long long>; using Z1 = function<Z2()>; // powers of 10 array constexpr auto pow10 = [] { array <long long, 19> n {1}; for (int j{0}, i{1}; i < 19; j = i++) n[i] = n[j] * 10; return n; } (); long long acc, l; bool izRev(int n, unsigned long long i, unsigned long long g) {
return (i / pow10[n - 1] != g % 10) ? false : n < 2 ? true : izRev(n - 1, i % pow10[n - 1], g / 10);
} const Z1 fG(Z1 n, int start, int end, int reset, const long long step, long long &l) {
return [n, i{step * start}, g{step * end}, e{step * reset}, &l, step] () mutable {
while (i<g){i+=step; return Z2(l+=step);}
l-=g-(i=e); return n();};
} struct nLH {
vector<unsigned long long>even{}, odd{};
nLH(const Z1 a, const vector<long long> b, long long llim){while (auto i = a()) for (auto ng : b)
if(ng>0 | *i>llim){unsigned long long sq{ng+ *i}, r{sqrt(sq)}; if (r*r == sq) ng&1 ? odd.push_back(sq) : even.push_back(sq);}}
}; const double fac = 3.94; const int mbs = (int)sqrt(fac * pow10[9]), mbt = (int)sqrt(fac * fac * pow10[9]) >> 3; const bitset<100000>bs {[]{bitset<100000>n{false}; for(int g{3};g<mbs;++g) n[(g*g)%100000]=true; return n;}()}; constexpr array<const int, 7>li{1,3,0,0,1,1,1},lin{0,-7,0,0,-8,-3,-9},lig{0,9,0,0,8,7,9},lil{0,2,0,0,2,10,2}; const nLH makeL(const int n){
constexpr int r{9}; acc=0; Z1 g{[]{return Z2{};}}; int s{-r}, q{(n>11)*5}; vector<long long> w{};
for (int i{1};i<n/2-q+1;++i){l=pow10[n-i-q]-pow10[i+q-1]; s-=i==n/2-q; g=fG(g,s,r,-r,l,acc+=l*s);}
if(q){long long g0{0}, g1{0}, g2{0}, g3{0}, g4{0}, l3{pow10[n-5]}; while (g0<7){const long long g{-10000*g4-1000*g3-100*g2-10*g1-g0};
if (bs[(g+1000000000000LL)%100000]) w.push_back(l3*(g4+g3*10+g2*100+g1*1000+g0*10000)+g);
if(g4<r) ++g4; else{g4= -r; if(g3<r) ++g3; else{g3= -r; if(g2<r) ++g2; else{g2= -r; if(g1<lig[g0]) g1+=lil[g0]; else {g0+=li[g0];g1=lin[g0];}}}}}}
return q ? nLH(g,w,0) : nLH(g,{0},0);
} const bitset<100000>bt {[]{bitset<100000>n{false}; for(int g{11};g<mbt;++g) n[(g*g)%100000]=true; return n;}()}; constexpr array<const int, 17>lu{0,0,0,0,2,0,4,0,0,0,1,4,0,0,0,1,1},lun{0,0,0,0,0,0,1,0,0,0,9,1,0,0,0,1,0},lug{0,0,0,0,18,0,17,0,0,0,9,17,0,0,0,11,18},lul{0,0,0,0,2,0,2,0,0,0,0,2,0,0,0,10,2}; const nLH makeH(const int n){
acc= -pow10[n>>1]-pow10[(n-1)>>1]; Z1 g{[]{ return Z2{};}}; int q{(n>11)*5}; vector<long long> w {};
for (int i{1}; i<(n>>1)-q+1; ++i) g = fG(g,0,18,0,pow10[n-i-q]+pow10[i+q-1], acc);
if (n & 1){l=pow10[n>>1]<<1; g=fG(g,0,9,0,l,acc+=l);}
if(q){long long g0{4}, g1{0}, g2{0}, g3{0}, g4{0},l3{pow10[n-5]}; while (g0<17){const long long g{g4*10000+g3*1000+g2*100+g1*10+g0};
if (bt[g%100000]) w.push_back(l3*(g4+g3*10+g2*100+g1*1000+g0*10000)+g);
if (g4<18) ++g4; else{g4=0; if(g3<18) ++g3; else{g3=0; if(g2<18) ++g2; else{g2=0; if(g1<lug[g0]) g1+=lul[g0]; else{g0+=lu[g0];g1=lun[g0];}}}}}}
return q ? nLH(g,w,0) : nLH(g,{0},pow10[n-1]<<2);
}
- include <chrono>
using namespace chrono; using VU = vector<unsigned long long>; using VS = vector<string>; template <typename T> // concatenates vectors vector<T>& operator +=(vector<T>& v, const vector<T>& w) { v.insert(v.end(), w.begin(), w.end()); return v; } int c{0}; // solution counter auto st{steady_clock::now()}, st0{st}, tmp{st}; // for determining elasped time // formats elasped time string dFmt(duration<double> et, int digs) {
string res{""}; double dt{et.count()};
if (dt > 60.0) { int m = (int)(dt / 60.0); dt -= m * 60.0; res = to_string(m) + "m"; }
res += to_string(dt); return res.substr(0, digs - 1) + 's';
} // combines list of square differences with list of square sums, reports compatible results VS dump(int nd, VU lo, VU hi) {
VS res {};
for (auto l : lo) for (auto h : hi) {
auto r { (h - l) >> 1 }, z { h - r };
if (izRev(nd, r, z)) {
char buf[99]; sprintf(buf, "%20llu %11lu %10lu", z, (long long)sqrt(h), (long long)sqrt(l));
res.push_back(buf); } } return res;
} // reports one block of digits void doOne(int n, nLH L, nLH H) {
VS lines = dump(n, L.even, H.even); lines += dump(n, L.odd , H.odd); sort(lines.begin(), lines.end());
duration<double> tet = (tmp = steady_clock::now()) - st; int ls = lines.size();
if (ls-- > 0)
for (int i{0}; i <= ls; ++i)
printf("%3d %s%s", ++c, lines[i].c_str(), i == ls ? "" : "\n");
else printf("%s", string(47, ' ').c_str());
printf(" %2d: %s %s\n", n, dFmt(tmp - st0, 8).c_str(), dFmt(tet, 8).c_str()); st0 = tmp;
} void Rare(int n) { doOne(n, makeL(n), makeH(n)); } int main(int argc, char *argv[]) {
int max{argc > 1 ? stoi(argv[1]) : 19}; if (max < 2) max = 2; if (max > 19 ) max = 19;
printf("%4s %19s %11s %10s %5s %11s %9s\n", "nth", "forward", "rt.sum", "rt.diff", "digs", "block.et", "total.et");
for (int nd{2}; nd <= max; ++nd) Rare(nd);
} </lang>
- Output:
Processor Intel(R) Core(TM) i5-1035G1 CPU @ 1.00GHz
nth forward rt.sum rt.diff digs block.et total.et
1 65 11 3 2: 0.00003s 0.00003s
3: 0.00002s 0.00006s
4: 0.00001s 0.00008s
5: 0.00003s 0.00011s
2 621770 836 738 6: 0.00007s 0.00018s
7: 0.00035s 0.00054s
8: 0.00110s 0.00164s
3 281089082 23708 330 9: 0.00657s 0.00821s
4 2022652202 63602 300
5 2042832002 63602 6360 10: 0.02246s 0.03068s
11: 0.11082s 0.14151s
6 868591084757 1275175 333333
7 872546974178 1320616 32670
8 872568754178 1320616 33330 12: 0.00868s 0.15019s
9 6979302951885 3586209 1047717 13: 0.03915s 0.18935s
10 20313693904202 6368252 269730
11 20313839704202 6368252 270270
12 20331657922202 6368252 329670
13 20331875722202 6368252 330330
14 20333875702202 6368252 336330
15 40313893704200 6368252 6330336
16 40351893720200 6368252 6336336 14: 0.11688s 0.30624s
17 200142385731002 20006998 69300
18 204238494066002 20122102 1891560
19 221462345754122 21045662 69300
20 244062891224042 22011022 1908060
21 245518996076442 22140228 921030
22 248359494187442 22206778 1891560
23 403058392434500 20211202 19940514
24 441054594034340 22011022 19940514
25 816984566129618 40421606 250800 15: 0.69490s 1.00114s
26 2078311262161202 64030648 7529850
27 2133786945766212 65272218 2666730
28 2135568943984212 65272218 3267330
29 2135764587964212 65272218 3326670
30 2135786765764212 65272218 3333330
31 4135786945764210 65272218 63333336
32 6157577986646405 105849161 33333333
33 6889765708183410 83866464 82133718
34 8052956026592517 123312255 29999997
35 8052956206592517 123312255 30000003
36 8191154686620818 127950856 3299670
37 8191156864620818 127950856 3300330
38 8191376864400818 127950856 3366330
39 8650327689541457 127246955 33299667
40 8650349867341457 127246955 33300333 16: 2.18232s 3.18347s
41 22542040692914522 212329862 333300
42 67725910561765640 269040196 251135808
43 86965750494756968 417050956 33000 17: 13.1765s 16.3599s
44 225342456863243522 671330638 297000
45 225342458663243522 671330638 303000
46 225342478643243522 671330638 363000
47 284684666566486482 754565658 30000
48 284684868364486482 754565658 636000
49 297128548234950692 770186978 32697330
50 297128722852950692 770186978 32702670
51 297148324656930692 770186978 33296670
52 297148546434930692 770186978 33303330
53 497168548234910690 770186978 633363336
54 619431353040136925 1071943279 299667003
55 619631153042134925 1071943279 300333003
56 631688638047992345 1083968809 297302703
57 633288858025996145 1083968809 302637303
58 633488632647994145 1083968809 303296697
59 653488856225994125 1083968809 363303363
60 811865096390477018 1273828556 33030330
61 865721270017296468 1315452006 32071170
62 871975098681469178 1320582934 3303300
63 898907259301737498 1339270086 64576740 18: 41.6983s 58.0583s
64 2042401829204402402 2021001202 18915600
65 2060303819041450202 2020110202 199405140
66 2420424089100600242 2200110022 19080600
67 2551755006254571552 2259094848 693000
68 2702373360882732072 2324811012 693000
69 2825378427312735282 2377130742 2508000
70 6531727101458000045 3454234451 1063822617
71 6988066446726832640 2729551744 2554541088
72 8066308349502036608 4016542096 2508000
73 8197906905009010818 4046976144 133408770
74 8200756128308135597 4019461925 495417087
75 8320411466598809138 4079154376 36366330 19: 5m1.342s 5m59.40s
20+ digits
<lang cpp> // Rare Numbers : Nigel Galloway - December 20th., 2019
- include <iostream>
- include <functional>
- include <bitset>
- include <gmpxx.h>
using Z2=std::optional<long>; using Z1=std::function<Z2()>; constexpr std::array<const long,19> pow10{1,10,100,1000,10000,100000,1000000,10000000,100000000,1000000000,10000000000,100000000000,1000000000000,10000000000000,100000000000000,1000000000000000,10000000000000000,100000000000000000,1000000000000000000}; const bool izRev(const mpz_class n,const mpz_class i,const mpz_class g){return (i/n!=g%10)? false : (n<2)? true : izRev(n/10,i%n,g/10);} const Z1 fG(Z1 n,int start, int end,int reset,const long step,long &l){return ([n,i{step*start},g{step*end},e{step*reset},&l,step]()mutable{
while(i<g){l+=step; i+=step; return Z2(l);} i=e; l-=(g-e); return n();});}
struct nLH{
std::vector<mpz_class>even{};
std::vector<mpz_class>odd{};
nLH(std::pair<Z1,std::vector<std::pair<long,long>>> e){auto [n,g]=e; mpz_t w,l,y; mpz_inits(w,l,y,NULL); mpz_set_si(w,pow10[4]);
while (auto i=n()){for(auto [ng,gg]:g){if((ng>0)|(*i>0)){mpz_set_si(y,gg+*i); mpz_addmul_ui(y,w,ng);
if(mpz_perfect_square_p(y)) (gg%2==0)? even.push_back(mpz_class(y)) : odd.push_back(mpz_class(y));}}} mpz_clears(w,l,y,NULL);}
}; class Rare{
mpz_class r,z,p;
long acc{0};
const std::bitset<10000>bs;
const std::pair<Z1,std::vector<std::pair<long,long>>> makeL(const int n){ //std::cout<<"Making L"<<std::endl;
Z1 g[n/2-3]; g[0]=([]{return Z2{};});
for(int i{1};i<n/2-3;++i){int s{(i==n/2-4)? -10:-9}; long l=pow10[n-i-4]-pow10[i+3]; acc+=l*s; g[i]=fG(g[i-1],s,9,-9,l,acc);}
return {g[n/2-4],([g0{0},g1{0},g2{0},g3{0},l3{pow10[n-8]},l2{pow10[n-7]},l1{pow10[n-6]},l0{pow10[n-5]},this]()mutable{std::vector<std::pair<long,long>>w{}; while (g0<10){
long n{g3*l3+g2*l2+g1*l1+g0*l0}; long g{-1000*g3-100*g2-10*g1-g0}; if(g3<9) ++g3; else{g3=-9; if(g2<9) ++g2; else{g2=-9; if(g1<9) ++g1; else{g1=-9; ++g0;}}}
if (bs[(pow10[10]+g)%10000]) w.push_back({n,g});} return w;})()};}
const std::pair<Z1,std::vector<std::pair<long,long>>> makeH(const int n){ acc=-(pow10[n/2]+pow10[(n-1)/2]); //std::cout<<"Making H"<<std::endl;
Z1 g[(n+1)/2-3]; g[0]=([]{return Z2{};});
for(int i{1};i<n/2-3;++i) g[i]=fG(g[i-1],(i==(n+1)/2-3)? -1:0,18,0,pow10[n-i-4]+pow10[i+3],acc);
if(n%2==1) g[(n+1)/2-4]=fG(g[n/2-4],-1,9,0,2*pow10[n/2],acc);
return {g[(n+1)/2-4],([g0{1},g1{0},g2{0},g3{0},l3{pow10[n-8]},l2{pow10[n-7]},l1{pow10[n-6]},l0{pow10[n-5]},this]()mutable{std::vector<std::pair<long,long>>w{}; while (g0<17){
long n{g3*l3+g2*l2+g1*l1+g0*l0}; long g{g3*1000+g2*100+g1*10+g0}; if(g3<18) ++g3; else{g3=0; if(g2<18) ++g2; else{g2=0; if(g1<18) ++g1; else{g1=0; ++g0;}}}
if (bs[g%10000]) w.push_back({n,g});} return w;})()};}
const nLH L,H;
public: Rare(int n):L{makeL(n)},H{makeH(n)},bs{([]{std::bitset<10000>n{false}; for(int g{0};g<10000;++g) n[(g*g)%10000]=true; return n;})()}{
mpz_ui_pow_ui(p.get_mpz_t(),10,n-1);
std::cout<<"Rare "<<n<<std::endl;
for(auto l:L.even) for(auto h:H.even){r=(h-l)/2; z=h-r; if(izRev(p,r,z)) std::cout<<"n="<<z<<" r="<<r<<" n-r="<<l<<" n+r="<<h<<std::endl;}
for(auto l:L.odd) for(auto h:H.odd) {r=(h-l)/2; z=h-r; if(izRev(p,r,z)) std::cout<<"n="<<z<<" r="<<r<<" n-r="<<l<<" n+r="<<h<<std::endl;}
}}; int main(){
Rare(20);
} </lang>
- Output:
Rare 17 n=67725910561765640 r=4656716501952776 n-r=63069194059812864 n+r=72382627063718416 n=86965750494756968 r=86965749405756968 n-r=1089000000 n+r=173931499900513936 n=22542040692914522 r=22541929604024522 n-r=111088890000 n+r=45083970296939044 Rare 18 n=865721270017296468 r=864692710072127568 n-r=1028559945168900 n+r=1730413980089424036 n=297128548234950692 r=296059432845821792 n-r=1069115389128900 n+r=593187981080772484 n=297128722852950692 r=296059258227821792 n-r=1069464625128900 n+r=593187981080772484 n=898907259301737498 r=894737103952709898 n-r=4170155349027600 n+r=1793644363254447396 n=811865096390477018 r=810774093690568118 n-r=1091002699908900 n+r=1622639190081045136 n=284684666566486482 r=284684665666486482 n-r=900000000 n+r=569369332232972964 n=225342456863243522 r=225342368654243522 n-r=88209000000 n+r=450684825517487044 n=225342458663243522 r=225342366854243522 n-r=91809000000 n+r=450684825517487044 n=225342478643243522 r=225342346874243522 n-r=131769000000 n+r=450684825517487044 n=284684868364486482 r=284684463868486482 n-r=404496000000 n+r=569369332232972964 n=297148324656930692 r=296039656423841792 n-r=1108668233088900 n+r=593187981080772484 n=297148546434930692 r=296039434645841792 n-r=1109111789088900 n+r=593187981080772484 n=871975098681469178 r=871964186890579178 n-r=10911790890000 n+r=1743939285572048356 n=497168548234910690 r=96019432845861794 n-r=401149115389048896 n+r=593187981080772484 n=633488632647994145 r=541499746236884336 n-r=91988886411109809 n+r=1174988378884878481 n=631688638047992345 r=543299740836886136 n-r=88388897211106209 n+r=1174988378884878481 n=653488856225994125 r=521499522658884356 n-r=131989333567109769 n+r=1174988378884878481 n=633288858025996145 r=541699520858882336 n-r=91589337167113809 n+r=1174988378884878481 n=619631153042134925 r=529431240351136916 n-r=90199912690998009 n+r=1149062393393271841 n=619431353040136925 r=529631040353134916 n-r=89800312687002009 n+r=1149062393393271841 Rare 20 n=22134434735752443122 r=22134425753743443122 n-r=8982009000000 n+r=44268860489495886244 n=22134434753752443122 r=22134425735743443122 n-r=9018009000000 n+r=44268860489495886244 n=22134436953532443122 r=22134423535963443122 n-r=13417569000000 n+r=44268860489495886244 n=65459144877856561700 r=716565877844195456 n-r=64742579000012366244 n+r=66175710755700757156 n=22136414517954423122 r=22132445971541463122 n-r=3968546412960000 n+r=44268860489495886244 n=22136414971554423122 r=22132445517941463122 n-r=3969453612960000 n+r=44268860489495886244 n=22136456771730423122 r=22132403717765463122 n-r=4053053964960000 n+r=44268860489495886244 n=61952807156239928885 r=58882993265170825916 n-r=3069813891069102969 n+r=120835800421410754801 n=61999171315484316965 r=56961348451317199916 n-r=5037822864167117049 n+r=118960519766801516881
D
Scaled down from the full duration showed in the go example because I got impatient and have not spent time figuring out where the inefficeny is. <lang d>import std.algorithm; import std.array; import std.conv; import std.datetime.stopwatch; import std.math; import std.stdio;
struct Term {
ulong coeff; byte ix1, ix2;
}
enum maxDigits = 16;
ulong toUlong(byte[] digits, bool reverse) {
ulong sum = 0;
if (reverse) {
for (int i = digits.length - 1; i >= 0; --i) {
sum = sum * 10 + digits[i];
}
} else {
for (size_t i = 0; i < digits.length; ++i) {
sum = sum * 10 + digits[i];
}
}
return sum;
}
bool isSquare(ulong n) {
if ((0x202021202030213 & (1 << (n & 63))) != 0) {
auto root = cast(ulong)sqrt(cast(double)n);
return root * root == n;
}
return false;
}
byte[] seq(byte from, byte to, byte step) {
byte[] res;
for (auto i = from; i <= to; i += step) {
res ~= i;
}
return res;
}
string commatize(ulong n) {
auto s = n.to!string;
auto le = s.length;
for (int i = le - 3; i >= 1; i -= 3) {
s = s[0..i] ~ "," ~ s[i..$];
}
return s;
}
void main() {
auto sw = StopWatch(AutoStart.yes);
ulong pow = 1;
writeln("Aggregate timings to process all numbers up to:");
// terms of (n-r) expression for number of digits from 2 to maxDigits
Term[][] allTerms = uninitializedArray!(Term[][])(maxDigits - 1);
for (auto r = 2; r <= maxDigits; r++) {
Term[] terms;
pow *= 10;
ulong pow1 = pow;
ulong pow2 = 1;
byte i1 = 0;
byte i2 = cast(byte)(r - 1);
while (i1 < i2) {
terms ~= Term(pow1 - pow2, i1, i2);
pow1 /= 10;
pow2 *= 10;
i1++;
i2--;
}
allTerms[r - 2] = terms;
}
// map of first minus last digits for 'n' to pairs giving this value
byte[][][byte] fml = [
0: [[2, 2], [8, 8]],
1: [[6, 5], [8, 7]],
4: 4, 0,
6: [[6, 0], [8, 2]]
];
// map of other digit differences for 'n' to pairs giving this value
byte[][][byte] dmd;
for (byte i = 0; i < 100; i++) {
byte[] a = [i / 10, i % 10];
auto d = a[0] - a[1];
dmd[cast(byte)d] ~= a;
}
byte[] fl = [0, 1, 4, 6];
auto dl = seq(-9, 9, 1); // all differences
byte[] zl = [0]; // zero diferences only
auto el = seq(-8, 8, 2); // even differences only
auto ol = seq(-9, 9, 2); // odd differences only
auto il = seq(0, 9, 1);
ulong[] rares;
byte[][][] lists = uninitializedArray!(byte[][][])(4);
foreach (i, f; fl) {
lists[i] = f;
}
byte[] digits;
int count = 0;
// Recursive closure to generate (n+r) candidates from (n-r) candidates
// and hence find Rare numbers with a given number of digits.
void fnpr(byte[] cand, byte[] di, byte[][] dis, byte[][] indicies, ulong nmr, int nd, int level) {
if (level == dis.length) {
digits[indicies[0][0]] = fml[cand[0]][di[0]][0];
digits[indicies[0][1]] = fml[cand[0]][di[0]][1];
auto le = di.length;
if (nd % 2 == 1) {
le--;
digits[nd / 2] = di[le];
}
foreach (i, d; di[1..le]) {
digits[indicies[i + 1][0]] = dmd[cand[i + 1]][d][0];
digits[indicies[i + 1][1]] = dmd[cand[i + 1]][d][1];
}
auto r = toUlong(digits, true);
auto npr = nmr + 2 * r;
if (!isSquare(npr)) {
return;
}
count++;
writef(" R/N %2d:", count);
auto ms = sw.peek();
writef(" %9s", ms);
auto n = toUlong(digits, false);
writef(" (%s)\n", commatize(n));
rares ~= n;
} else {
foreach (num; dis[level]) {
di[level] = num;
fnpr(cand, di, dis, indicies, nmr, nd, level + 1);
}
}
}
// Recursive closure to generate (n-r) candidates with a given number of digits.
void fnmr(byte[] cand, byte[][] list, byte[][] indicies, int nd, int level) {
if (level == list.length) {
ulong nmr, nmr2;
foreach (i, t; allTerms[nd - 2]) {
if (cand[i] >= 0) {
nmr += t.coeff * cand[i];
} else {
nmr2 += t.coeff * -cast(int)(cand[i]);
if (nmr >= nmr2) {
nmr -= nmr2;
nmr2 = 0;
} else {
nmr2 -= nmr;
nmr = 0;
}
}
}
if (nmr2 >= nmr) {
return;
}
nmr -= nmr2;
if (!isSquare(nmr)) {
return;
}
byte[][] dis;
dis ~= seq(0, cast(byte)(fml[cand[0]].length - 1), 1);
for (auto i = 1; i < cand.length; i++) {
dis ~= seq(0, cast(byte)(dmd[cand[i]].length - 1), 1);
}
if (nd % 2 == 1) {
dis ~= il;
}
byte[] di = uninitializedArray!(byte[])(dis.length);
fnpr(cand, di, dis, indicies, nmr, nd, 0);
} else {
foreach (num; list[level]) {
cand[level] = num;
fnmr(cand, list, indicies, nd, level + 1);
}
}
}
for (int nd = 2; nd <= maxDigits; nd++) {
digits = uninitializedArray!(byte[])(nd);
if (nd == 4) {
lists[0] ~= zl;
lists[1] ~= ol;
lists[2] ~= el;
lists[3] ~= ol;
} else if (allTerms[nd - 2].length > lists[0].length) {
for (int i = 0; i < 4; i++) {
lists[i] ~= dl;
}
}
byte[][] indicies;
foreach (t; allTerms[nd - 2]) {
indicies ~= [t.ix1, t.ix2];
}
foreach (list; lists) {
byte[] cand = uninitializedArray!(byte[])(list.length);
fnmr(cand, list, indicies, nd, 0);
}
auto ms = sw.peek();
writefln(" %2d digits: %9s", nd, ms);
}
rares.sort;
writefln("\nThe rare numbers with up to %d digits are:", maxDigits);
foreach (i, rare; rares) {
writefln(" %2d: %25s", i + 1, commatize(rare));
}
}</lang>
- Output:
Aggregate timings to process all numbers up to:
R/N 1: 183 ╬╝s and 2 hnsecs (65)
2 digits: 193 ╬╝s and 8 hnsecs
3 digits: 301 ╬╝s and 3 hnsecs
4 digits: 380 ╬╝s and 1 hnsec
5 digits: 447 ╬╝s
R/N 2: 732 ╬╝s and 9 hnsecs (621,770)
6 digits: 767 ╬╝s and 1 hnsec
7 digits: 1 ms, 291 ╬╝s, and 5 hnsecs
8 digits: 5 ms, 602 ╬╝s, and 2 hnsecs
R/N 3: 5 ms, 900 ╬╝s, and 2 hnsecs (281,089,082)
9 digits: 7 ms, 537 ╬╝s, and 1 hnsec
R/N 4: 7 ms, 869 ╬╝s, and 5 hnsecs (2,022,652,202)
R/N 5: 32 ms, 826 ╬╝s, and 4 hnsecs (2,042,832,002)
10 digits: 96 ms, 422 ╬╝s, and 3 hnsecs
11 digits: 161 ms, 218 ╬╝s, and 4 hnsecs
R/N 6: 468 ms, 23 ╬╝s, and 9 hnsecs (872,546,974,178)
R/N 7: 506 ms, 702 ╬╝s, and 3 hnsecs (872,568,754,178)
R/N 8: 1 sec, 39 ms, 845 ╬╝s, and 6 hnsecs (868,591,084,757)
12 digits: 1 sec, 437 ms, 602 ╬╝s, and 8 hnsecs
R/N 9: 1 sec, 835 ms, 95 ╬╝s, and 6 hnsecs (6,979,302,951,885)
13 digits: 2 secs, 487 ms, 165 ╬╝s, and 9 hnsecs
R/N 10: 7 secs, 241 ms, 437 ╬╝s, and 1 hnsec (20,313,693,904,202)
R/N 11: 7 secs, 330 ms, 171 ╬╝s, and 2 hnsecs (20,313,839,704,202)
R/N 12: 9 secs, 290 ms, 907 ╬╝s, and 3 hnsecs (20,331,657,922,202)
R/N 13: 9 secs, 582 ms, 920 ╬╝s, and 5 hnsecs (20,331,875,722,202)
R/N 14: 10 secs, 383 ms, 769 ╬╝s, and 1 hnsec (20,333,875,702,202)
R/N 15: 25 secs, 835 ms, and 933 ╬╝s (40,313,893,704,200)
R/N 16: 26 secs, 14 ms, 774 ╬╝s, and 4 hnsecs (40,351,893,720,200)
14 digits: 30 secs, 110 ms, 971 ╬╝s, and 7 hnsecs
R/N 17: 30 secs, 216 ms, 437 ╬╝s, and 3 hnsecs (200,142,385,731,002)
R/N 18: 30 secs, 489 ms, 719 ╬╝s, and 2 hnsecs (221,462,345,754,122)
R/N 19: 34 secs, 83 ms, 642 ╬╝s, and 9 hnsecs (816,984,566,129,618)
R/N 20: 35 secs, 971 ms, 413 ╬╝s, and 3 hnsecs (245,518,996,076,442)
R/N 21: 36 secs, 250 ms, 787 ╬╝s, and 8 hnsecs (204,238,494,066,002)
R/N 22: 36 secs, 332 ms, 714 ╬╝s, and 2 hnsecs (248,359,494,187,442)
R/N 23: 36 secs, 696 ms, 902 ╬╝s, and 2 hnsecs (244,062,891,224,042)
R/N 24: 44 secs, 896 ms, and 665 ╬╝s (403,058,392,434,500)
R/N 25: 45 secs, 181 ms, 141 ╬╝s, and 5 hnsecs (441,054,594,034,340)
15 digits: 49 secs, 315 ms, 407 ╬╝s, and 4 hnsecs
R/N 26: 1 minute, 55 secs, 748 ms, 43 ╬╝s, and 4 hnsecs (2,133,786,945,766,212)
R/N 27: 2 minutes, 21 secs, 484 ms, 683 ╬╝s, and 7 hnsecs (2,135,568,943,984,212)
R/N 28: 2 minutes, 25 secs, 438 ms, 771 ╬╝s, and 7 hnsecs (8,191,154,686,620,818)
R/N 29: 2 minutes, 28 secs, 883 ms, 999 ╬╝s, and 6 hnsecs (8,191,156,864,620,818)
R/N 30: 2 minutes, 30 secs, 410 ms, and 831 ╬╝s (2,135,764,587,964,212)
R/N 31: 2 minutes, 32 secs, 594 ms, 842 ╬╝s, and 1 hnsec (2,135,786,765,764,212)
R/N 32: 2 minutes, 37 secs, 880 ms, 100 ╬╝s, and 5 hnsecs (8,191,376,864,400,818)
R/N 33: 2 minutes, 55 secs, 943 ms, 190 ╬╝s, and 5 hnsecs (2,078,311,262,161,202)
R/N 34: 3 minutes, 49 secs, 750 ms, 39 ╬╝s, and 5 hnsecs (8,052,956,026,592,517)
R/N 35: 3 minutes, 55 secs, 554 ms, 720 ╬╝s, and 1 hnsec (8,052,956,206,592,517)
R/N 36: 4 minutes, 41 secs, 59 ms, 309 ╬╝s, and 4 hnsecs (8,650,327,689,541,457)
R/N 37: 4 minutes, 43 secs, 951 ms, and 206 ╬╝s (8,650,349,867,341,457)
R/N 38: 4 minutes, 46 secs, 85 ms, 249 ╬╝s, and 7 hnsecs (6,157,577,986,646,405)
R/N 39: 5 minutes, 59 secs, 80 ms, 228 ╬╝s, and 5 hnsecs (4,135,786,945,764,210)
R/N 40: 7 minutes, 10 secs, 573 ms, 592 ╬╝s, and 2 hnsecs (6,889,765,708,183,410)
16 digits: 7 minutes, 16 secs, 827 ms, 76 ╬╝s, and 4 hnsecs
The rare numbers with up to 16 digits are:
1: 65
2: 621,770
3: 281,089,082
4: 2,022,652,202
5: 2,042,832,002
6: 868,591,084,757
7: 872,546,974,178
8: 872,568,754,178
9: 6,979,302,951,885
10: 20,313,693,904,202
11: 20,313,839,704,202
12: 20,331,657,922,202
13: 20,331,875,722,202
14: 20,333,875,702,202
15: 40,313,893,704,200
16: 40,351,893,720,200
17: 200,142,385,731,002
18: 204,238,494,066,002
19: 221,462,345,754,122
20: 244,062,891,224,042
21: 245,518,996,076,442
22: 248,359,494,187,442
23: 403,058,392,434,500
24: 441,054,594,034,340
25: 816,984,566,129,618
26: 2,078,311,262,161,202
27: 2,133,786,945,766,212
28: 2,135,568,943,984,212
29: 2,135,764,587,964,212
30: 2,135,786,765,764,212
31: 4,135,786,945,764,210
32: 6,157,577,986,646,405
33: 6,889,765,708,183,410
34: 8,052,956,026,592,517
35: 8,052,956,206,592,517
36: 8,191,154,686,620,818
37: 8,191,156,864,620,818
38: 8,191,376,864,400,818
39: 8,650,327,689,541,457
40: 8,650,349,867,341,457
F#
The Function
This solution demonstrates the concept described in Talk:Rare_numbers#30_mins_not_30_years. It doesn't use Cartesian_product_of_two_or_more_lists#Extra_Credit <lang fsharp> // Find all Rare numbers with a digits. Nigel Galloway: September 18th., 2019. let rareNums a=
let tN=set[1L;4L;5L;6L;9L]
let izPS g=let n=(float>>sqrt>>int64)g in n*n=g
let n=[for n in [0..a/2-1] do yield ((pown 10L (a-n-1))-(pown 10L n))]|>List.rev
let rec fN i g e=seq{match e with 0->yield g |e->for n in i do yield! fN [-9L..9L] (n::g) (e-1)}|>Seq.filter(fun g->let g=Seq.map2(*) n g|>Seq.sum in g>0L && izPS g)
let rec fG n i g e l=seq{
match l with
h::t->for l in max 0L (0L-h)..min 9L (9L-h) do if e>1L||l=0L||tN.Contains((2L*l+h)%10L) then yield! fG (n+l*e+(l+h)*g) (i+l*g+(l+h)*e) (g/10L) (e*10L) t
|_->if n>(pown 10L (a-1)) then for l in (if a%2=0 then [0L] else [0L..9L]) do let g=l*(pown 10L (a/2)) in if izPS (n+i+2L*g) then yield (i+g,n+g)}
fN [0L..9L] [] (a/2) |> Seq.collect(List.rev >> fG 0L 0L (pown 10L (a-1)) 1L)
</lang>
43 down
<lang fsharp> let test n=
let t = System.Diagnostics.Stopwatch.StartNew() for n in (rareNums n) do printfn "%A" n t.Stop() printfn "Elapsed Time: %d ms for length %d" t.ElapsedMilliseconds n
[2..17] |> Seq.iter test </lang>
- Output:
(56L, 65L) Elapsed Time: 31 ms for length 2 Elapsed Time: 0 ms for length 3 Elapsed Time: 0 ms for length 4 Elapsed Time: 0 ms for length 5 (77126L, 621770L) Elapsed Time: 6 ms for length 6 Elapsed Time: 6 ms for length 7 Elapsed Time: 113 ms for length 8 (280980182L, 281089082L) Elapsed Time: 72 ms for length 9 (2022562202L, 2022652202L) (2002382402L, 2042832002L) Elapsed Time: 1525 ms for length 10 Elapsed Time: 1351 ms for length 11 (871479645278L, 872546974178L) (871457865278L, 872568754178L) (757480195868L, 868591084757L) Elapsed Time: 27990 ms for length 12 (5881592039796L, 6979302951885L) Elapsed Time: 26051 ms for length 13 (20240939631302L, 20313693904202L) (20240793831302L, 20313839704202L) (20222975613302L, 20331657922202L) (20222757813302L, 20331875722202L) (20220757833302L, 20333875702202L) (240739831304L, 40313893704200L) (202739815304L, 40351893720200L) Elapsed Time: 552922 ms for length 14 (200137583241002L, 200142385731002L) (221457543264122L, 221462345754122L) (816921665489618L, 816984566129618L) (244670699815542L, 245518996076442L) (200660494832402L, 204238494066002L) (244781494953842L, 248359494187442L) (240422198260442L, 244062891224042L) (5434293850304L, 403058392434500L) (43430495450144L, 441054594034340L) Elapsed Time: 512282 ms for length 15 (2126675496873312L, 2133786945766212L) (2124893498655312L, 2135568943984212L) (8180266864511918L, 8191154686620818L) (8180264686511918L, 8191156864620818L) (2124697854675312L, 2135764587964212L) (2124675676875312L, 2135786765764212L) (8180044686731918L, 8191376864400818L) (2021612621138702L, 2078311262161202L) (7152956206592508L, 8052956026592517L) (7152956026592508L, 8052956206592517L) (7541459867230568L, 8650327689541457L) (7541437689430568L, 8650349867341457L) (5046466897757516L, 6157577986646405L) (124675496875314L, 4135786945764210L) (143818075679886L, 6889765708183410L) Elapsed Time: 11568713 ms for length 16 (86965749405756968L, 86965750494756968L) (22541929604024522L, 22542040692914522L) (4656716501952776L, 67725910561765640L) Elapsed Time: 11275839 ms for length 17
FreeBASIC
Made some changes and added a simple test to speed things up. Results in about 1 minute.
<lang freebasic>Function revn(n As ULongInt, nd As ULongInt) As ULongInt
Dim As ULongInt r
For i As UInteger = 1 To nd
r = r * 10 + n Mod 10
n = n \ 10
Next i
Return r
End Function
Dim As UInteger nd = 2, count, lim = 90, n = 20
Do
n += 1
Dim As ULongInt r = revn(n,nd)
If r < n Then
Dim As ULongInt s = n + r, d = n - r
If nd And 1 Then
If d Mod 1089 <> 0 Then GoTo jump
Else
If s Mod 121 <> 0 Then GoTo jump
End If
If Frac(Sqr(s)) = 0 And Frac(Sqr(d)) = 0 Then
count += 1
Print count; ": "; n
If count = 5 Then Exit Do : End If
End If
End If
jump:
If n = lim Then
lim = lim * 10
nd += 1
n = (lim \ 9) * 2
End If
Loop
Print Print "Done" Sleep</lang>
- Output:
1: 65 2: 621770 3: 281089082 4: 2022652202 5: 2042832002
Go
Traditional
This uses many of the hints within Shyam Sunder Gupta's webpage combined with Nigel Galloway's general approach (see Talk page) of working from (n-r) and deducing the Rare numbers with various numbers of digits from there.
As the algorithm used does not generate the Rare numbers in order, a sorted list is also printed. <lang go>package main
import (
"fmt" "math" "sort" "time"
)
type term struct {
coeff uint64 ix1, ix2 int8
}
const maxDigits = 19
func toUint64(digits []int8, reverse bool) uint64 {
sum := uint64(0)
if !reverse {
for i := 0; i < len(digits); i++ {
sum = sum*10 + uint64(digits[i])
}
} else {
for i := len(digits) - 1; i >= 0; i-- {
sum = sum*10 + uint64(digits[i])
}
}
return sum
}
func isSquare(n uint64) bool {
if 0x202021202030213&(1<<(n&63)) != 0 {
root := uint64(math.Sqrt(float64(n)))
return root*root == n
}
return false
}
func seq(from, to, step int8) []int8 {
var res []int8
for i := from; i <= to; i += step {
res = append(res, i)
}
return res
}
func commatize(n uint64) string {
s := fmt.Sprintf("%d", n)
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
return s
}
func main() {
start := time.Now()
pow := uint64(1)
fmt.Println("Aggregate timings to process all numbers up to:")
// terms of (n-r) expression for number of digits from 2 to maxDigits
allTerms := make([][]term, maxDigits-1)
for r := 2; r <= maxDigits; r++ {
var terms []term
pow *= 10
pow1, pow2 := pow, uint64(1)
for i1, i2 := int8(0), int8(r-1); i1 < i2; i1, i2 = i1+1, i2-1 {
terms = append(terms, term{pow1 - pow2, i1, i2})
pow1 /= 10
pow2 *= 10
}
allTerms[r-2] = terms
}
// map of first minus last digits for 'n' to pairs giving this value
fml := map[int8][][]int8{
0: {{2, 2}, {8, 8}},
1: {{6, 5}, {8, 7}},
4: Template:4, 0,
6: {{6, 0}, {8, 2}},
}
// map of other digit differences for 'n' to pairs giving this value
dmd := make(map[int8][][]int8)
for i := int8(0); i < 100; i++ {
a := []int8{i / 10, i % 10}
d := a[0] - a[1]
dmd[d] = append(dmd[d], a)
}
fl := []int8{0, 1, 4, 6}
dl := seq(-9, 9, 1) // all differences
zl := []int8{0} // zero differences only
el := seq(-8, 8, 2) // even differences only
ol := seq(-9, 9, 2) // odd differences only
il := seq(0, 9, 1)
var rares []uint64
lists := make([][][]int8, 4)
for i, f := range fl {
lists[i] = [][]int8Template:F
}
var digits []int8
count := 0
// Recursive closure to generate (n+r) candidates from (n-r) candidates
// and hence find Rare numbers with a given number of digits.
var fnpr func(cand, di []int8, dis [][]int8, indices [][2]int8, nmr uint64, nd, level int)
fnpr = func(cand, di []int8, dis [][]int8, indices [][2]int8, nmr uint64, nd, level int) {
if level == len(dis) {
digits[indices[0][0]] = fml[cand[0]][di[0]][0]
digits[indices[0][1]] = fml[cand[0]][di[0]][1]
le := len(di)
if nd%2 == 1 {
le--
digits[nd/2] = di[le]
}
for i, d := range di[1:le] {
digits[indices[i+1][0]] = dmd[cand[i+1]][d][0]
digits[indices[i+1][1]] = dmd[cand[i+1]][d][1]
}
r := toUint64(digits, true)
npr := nmr + 2*r
if !isSquare(npr) {
return
}
count++
fmt.Printf(" R/N %2d:", count)
ms := uint64(time.Since(start).Milliseconds())
fmt.Printf(" %9s ms", commatize(ms))
n := toUint64(digits, false)
fmt.Printf(" (%s)\n", commatize(n))
rares = append(rares, n)
} else {
for _, num := range dis[level] {
di[level] = num
fnpr(cand, di, dis, indices, nmr, nd, level+1)
}
}
}
// Recursive closure to generate (n-r) candidates with a given number of digits.
var fnmr func(cand []int8, list [][]int8, indices [][2]int8, nd, level int)
fnmr = func(cand []int8, list [][]int8, indices [][2]int8, nd, level int) {
if level == len(list) {
var nmr, nmr2 uint64
for i, t := range allTerms[nd-2] {
if cand[i] >= 0 {
nmr += t.coeff * uint64(cand[i])
} else {
nmr2 += t.coeff * uint64(-cand[i])
if nmr >= nmr2 {
nmr -= nmr2
nmr2 = 0
} else {
nmr2 -= nmr
nmr = 0
}
}
}
if nmr2 >= nmr {
return
}
nmr -= nmr2
if !isSquare(nmr) {
return
}
var dis [][]int8
dis = append(dis, seq(0, int8(len(fml[cand[0]]))-1, 1))
for i := 1; i < len(cand); i++ {
dis = append(dis, seq(0, int8(len(dmd[cand[i]]))-1, 1))
}
if nd%2 == 1 {
dis = append(dis, il)
}
di := make([]int8, len(dis))
fnpr(cand, di, dis, indices, nmr, nd, 0)
} else {
for _, num := range list[level] {
cand[level] = num
fnmr(cand, list, indices, nd, level+1)
}
}
}
for nd := 2; nd <= maxDigits; nd++ {
digits = make([]int8, nd)
if nd == 4 {
lists[0] = append(lists[0], zl)
lists[1] = append(lists[1], ol)
lists[2] = append(lists[2], el)
lists[3] = append(lists[3], ol)
} else if len(allTerms[nd-2]) > len(lists[0]) {
for i := 0; i < 4; i++ {
lists[i] = append(lists[i], dl)
}
}
var indices [][2]int8
for _, t := range allTerms[nd-2] {
indices = append(indices, [2]int8{t.ix1, t.ix2})
}
for _, list := range lists {
cand := make([]int8, len(list))
fnmr(cand, list, indices, nd, 0)
}
ms := uint64(time.Since(start).Milliseconds())
fmt.Printf(" %2d digits: %9s ms\n", nd, commatize(ms))
}
sort.Slice(rares, func(i, j int) bool { return rares[i] < rares[j] })
fmt.Printf("\nThe rare numbers with up to %d digits are:\n", maxDigits)
for i, rare := range rares {
fmt.Printf(" %2d: %25s\n", i+1, commatize(rare))
}
}</lang>
- Output:
Timings are for an Intel Core i7-8565U machine with 32GB RAM running Go 1.13.3 on Ubuntu 18.04.
Aggregate timings to process all numbers up to:
R/N 1: 0 ms (65)
2 digits: 0 ms
3 digits: 0 ms
4 digits: 0 ms
5 digits: 0 ms
R/N 2: 0 ms (621,770)
6 digits: 0 ms
7 digits: 0 ms
8 digits: 3 ms
R/N 3: 3 ms (281,089,082)
9 digits: 4 ms
R/N 4: 5 ms (2,022,652,202)
R/N 5: 31 ms (2,042,832,002)
10 digits: 71 ms
11 digits: 109 ms
R/N 6: 328 ms (872,546,974,178)
R/N 7: 355 ms (872,568,754,178)
R/N 8: 697 ms (868,591,084,757)
12 digits: 848 ms
R/N 9: 1,094 ms (6,979,302,951,885)
13 digits: 1,406 ms
R/N 10: 5,121 ms (20,313,693,904,202)
R/N 11: 5,187 ms (20,313,839,704,202)
R/N 12: 6,673 ms (20,331,657,922,202)
R/N 13: 6,887 ms (20,331,875,722,202)
R/N 14: 7,479 ms (20,333,875,702,202)
R/N 15: 17,112 ms (40,313,893,704,200)
R/N 16: 17,248 ms (40,351,893,720,200)
14 digits: 18,338 ms
R/N 17: 18,356 ms (200,142,385,731,002)
R/N 18: 18,560 ms (221,462,345,754,122)
R/N 19: 21,181 ms (816,984,566,129,618)
R/N 20: 22,491 ms (245,518,996,076,442)
R/N 21: 22,674 ms (204,238,494,066,002)
R/N 22: 22,734 ms (248,359,494,187,442)
R/N 23: 22,994 ms (244,062,891,224,042)
R/N 24: 26,868 ms (403,058,392,434,500)
R/N 25: 27,063 ms (441,054,594,034,340)
15 digits: 28,087 ms
R/N 26: 74,120 ms (2,133,786,945,766,212)
R/N 27: 92,245 ms (2,135,568,943,984,212)
R/N 28: 94,972 ms (8,191,154,686,620,818)
R/N 29: 97,313 ms (8,191,156,864,620,818)
R/N 30: 98,361 ms (2,135,764,587,964,212)
R/N 31: 99,971 ms (2,135,786,765,764,212)
R/N 32: 103,603 ms (8,191,376,864,400,818)
R/N 33: 115,711 ms (2,078,311,262,161,202)
R/N 34: 140,972 ms (8,052,956,026,592,517)
R/N 35: 145,099 ms (8,052,956,206,592,517)
R/N 36: 175,023 ms (8,650,327,689,541,457)
R/N 37: 177,145 ms (8,650,349,867,341,457)
R/N 38: 178,693 ms (6,157,577,986,646,405)
R/N 39: 205,564 ms (4,135,786,945,764,210)
R/N 40: 220,480 ms (6,889,765,708,183,410)
16 digits: 221,485 ms
R/N 41: 226,520 ms (86,965,750,494,756,968)
R/N 42: 227,431 ms (22,542,040,692,914,522)
R/N 43: 345,314 ms (67,725,910,561,765,640)
17 digits: 354,815 ms
R/N 44: 387,879 ms (284,684,666,566,486,482)
R/N 45: 510,788 ms (225,342,456,863,243,522)
R/N 46: 556,239 ms (225,342,458,663,243,522)
R/N 47: 652,051 ms (225,342,478,643,243,522)
R/N 48: 718,148 ms (284,684,868,364,486,482)
R/N 49: 1,095,093 ms (871,975,098,681,469,178)
R/N 50: 1,785,243 ms (865,721,270,017,296,468)
R/N 51: 1,800,799 ms (297,128,548,234,950,692)
R/N 52: 1,809,196 ms (297,128,722,852,950,692)
R/N 53: 1,913,085 ms (811,865,096,390,477,018)
R/N 54: 1,965,104 ms (297,148,324,656,930,692)
R/N 55: 1,990,018 ms (297,148,546,434,930,692)
R/N 56: 2,296,972 ms (898,907,259,301,737,498)
R/N 57: 2,691,110 ms (631,688,638,047,992,345)
R/N 58: 2,716,487 ms (619,431,353,040,136,925)
R/N 59: 2,984,451 ms (619,631,153,042,134,925)
R/N 60: 3,047,183 ms (633,288,858,025,996,145)
R/N 61: 3,115,724 ms (633,488,632,647,994,145)
R/N 62: 3,978,143 ms (653,488,856,225,994,125)
R/N 63: 4,255,985 ms (497,168,548,234,910,690)
18 digits: 4,531,846 ms
R/N 64: 4,606,094 ms (2,551,755,006,254,571,552)
R/N 65: 4,624,539 ms (2,702,373,360,882,732,072)
R/N 66: 4,873,160 ms (2,825,378,427,312,735,282)
R/N 67: 4,893,810 ms (8,066,308,349,502,036,608)
R/N 68: 5,109,513 ms (2,042,401,829,204,402,402)
R/N 69: 5,152,863 ms (2,420,424,089,100,600,242)
R/N 70: 5,263,434 ms (8,320,411,466,598,809,138)
R/N 71: 5,558,356 ms (8,197,906,905,009,010,818)
R/N 72: 5,586,801 ms (2,060,303,819,041,450,202)
R/N 73: 5,763,382 ms (8,200,756,128,308,135,597)
R/N 74: 6,008,475 ms (6,531,727,101,458,000,045)
R/N 75: 6,543,047 ms (6,988,066,446,726,832,640)
19 digits: 6,609,905 ms
The rare numbers with up to 19 digits are:
1: 65
2: 621,770
3: 281,089,082
4: 2,022,652,202
5: 2,042,832,002
6: 868,591,084,757
7: 872,546,974,178
8: 872,568,754,178
9: 6,979,302,951,885
10: 20,313,693,904,202
11: 20,313,839,704,202
12: 20,331,657,922,202
13: 20,331,875,722,202
14: 20,333,875,702,202
15: 40,313,893,704,200
16: 40,351,893,720,200
17: 200,142,385,731,002
18: 204,238,494,066,002
19: 221,462,345,754,122
20: 244,062,891,224,042
21: 245,518,996,076,442
22: 248,359,494,187,442
23: 403,058,392,434,500
24: 441,054,594,034,340
25: 816,984,566,129,618
26: 2,078,311,262,161,202
27: 2,133,786,945,766,212
28: 2,135,568,943,984,212
29: 2,135,764,587,964,212
30: 2,135,786,765,764,212
31: 4,135,786,945,764,210
32: 6,157,577,986,646,405
33: 6,889,765,708,183,410
34: 8,052,956,026,592,517
35: 8,052,956,206,592,517
36: 8,191,154,686,620,818
37: 8,191,156,864,620,818
38: 8,191,376,864,400,818
39: 8,650,327,689,541,457
40: 8,650,349,867,341,457
41: 22,542,040,692,914,522
42: 67,725,910,561,765,640
43: 86,965,750,494,756,968
44: 225,342,456,863,243,522
45: 225,342,458,663,243,522
46: 225,342,478,643,243,522
47: 284,684,666,566,486,482
48: 284,684,868,364,486,482
49: 297,128,548,234,950,692
50: 297,128,722,852,950,692
51: 297,148,324,656,930,692
52: 297,148,546,434,930,692
53: 497,168,548,234,910,690
54: 619,431,353,040,136,925
55: 619,631,153,042,134,925
56: 631,688,638,047,992,345
57: 633,288,858,025,996,145
58: 633,488,632,647,994,145
59: 653,488,856,225,994,125
60: 811,865,096,390,477,018
61: 865,721,270,017,296,468
62: 871,975,098,681,469,178
63: 898,907,259,301,737,498
64: 2,042,401,829,204,402,402
65: 2,060,303,819,041,450,202
66: 2,420,424,089,100,600,242
67: 2,551,755,006,254,571,552
68: 2,702,373,360,882,732,072
69: 2,825,378,427,312,735,282
70: 6,531,727,101,458,000,045
71: 6,988,066,446,726,832,640
72: 8,066,308,349,502,036,608
73: 8,197,906,905,009,010,818
74: 8,200,756,128,308,135,597
75: 8,320,411,466,598,809,138
Quicker
Twice as quick as the first Go version though, despite being a faithful translation, a little slower than the C# and VB.NET versions. <lang go>package main
import (
"fmt" "math" "sort" "time"
)
type llst = [][]int
var (
d []int // permutation working slice drar [19]int // digital root lookup array dac []int // running digital root slice p [20]int64 // powers of 10 ac []int64 // accumulator slice pp []int64 // coefficient slice that combines with digits of working slice sr []int64 // temporary list of squares used for building
)
var (
odd = false // flag for odd number of digits sum int64 // calculated sum of terms (square candidate) rt int64 // root of sum cn = 0 // solution counter nd = 2 // number of digits nd1 = nd - 1 // 'nd' helper ln int // previous value of 'n' (in recurse()) dl int // length of 'd' slice
)
var (
tlo = []int{0, 1, 4, 5, 6} // primary differences starting point
all = seq(-9, 9, 1) // all possible differences
odl = seq(-9, 9, 2) // odd possible differences
evl = seq(-8, 8, 2) // even possible differences
thi = []int{4, 5, 6, 9, 10, 11, 14, 15, 16} // primary sums starting point
alh = seq(0, 18, 1) // all possible sums
odh = seq(1, 17, 2) // odd possible sums
evh = seq(0, 18, 2) // even possible sums
ten = seq(0, 9, 1) // used for odd number of digits
z = seq(0, 0, 1) // no difference, avoids generating a bunch of negative square candidates
t7 = []int{-3, 7} // shortcut for low 5
nin = []int{9} // shortcut for hi 10
tn = []int{10} // shortcut for hi 0 (unused, unneeded)
t12 = []int{2, 12} // shortcut for hi 5
o11 = []int{1, 11} // shortcut for hi 15
pos = []int{0, 1, 4, 5, 6, 9} // shortcut for 2nd lo 0
)
var (
lul = llst{z, odl, nil, nil, evl, t7, odl} // shortcut lookup lo primary
luh = llst{tn, evh, nil, nil, evh, t12, odh, nil, nil, evh, nin, odh, nil, nil,
odh, o11, evh} // shortcut lookup hi primary
l2l = llst{pos, nil, nil, nil, all, nil, all} // shortcut lookup lo secondary
l2h = llst{nil, nil, nil, nil, alh, nil, alh, nil, nil, nil, alh, nil, nil, nil,
alh, nil, alh} // shortcut lookup hi secondary
lu, l2 llst // ditto
chTen = llst{{0, 2, 5, 8, 9}, {0, 3, 4, 6, 9}, {1, 4, 7, 8}, {2, 3, 5, 8},
{0, 3, 6, 7, 9}, {1, 2, 4, 7}, {2, 5, 6, 8}, {0, 1, 3, 6, 9}, {1, 4, 5, 7}}
chAH = llst{{0, 2, 5, 8, 9, 11, 14, 17, 18}, {0, 3, 4, 6, 9, 12, 13, 15, 18}, {1, 4, 7, 8, 10, 13, 16, 17},
{2, 3, 5, 8, 11, 12, 14, 17}, {0, 3, 6, 7, 9, 12, 15, 16, 18}, {1, 2, 4, 7, 10, 11, 13, 16},
{2, 5, 6, 8, 11, 14, 15, 17}, {0, 1, 3, 6, 9, 10, 12, 15, 18}, {1, 4, 5, 7, 10, 13, 14, 16}}
)
// Returns a sequence of integers. func seq(f, t, s int) []int {
r := make([]int, (t-f)/s+1)
for i := 0; i < len(r); i, f = i+1, f+s {
r[i] = f
}
return r
}
// Returns Integer Square Root. func isr(s int64) int64 {
return int64(math.Sqrt(float64(s)))
}
// Recursively determines whether 'r' is the reverse of 'f'. func isRev(nd int, f, r int64) bool {
nd--
if f/p[nd] != r%10 {
return false
}
if nd < 1 {
return true
}
return isRev(nd, f%p[nd], r/10)
}
// Recursive function to evaluate the permutations, no shortcuts. func recurseLE5(lst llst, lv int) {
if lv == dl { // check if on last stage of permutation
sum = ac[lv-1]
if sum > 0 {
rt = int64(math.Sqrt(float64(sum)))
if rt*rt == sum { // test accumulated sum, append to result if square
sr = append(sr, sum)
}
}
} else {
for _, n := range lst[lv] { // set up next permutation
d[lv] = n
if lv == 0 {
ac[0] = pp[0] * int64(n)
} else {
ac[lv] = ac[lv-1] + pp[lv]*int64(n) // update accumulated sum
}
recurseLE5(lst, lv+1) // recursively call next level
}
}
}
// Recursive function to evaluate the hi permutations, shortcuts added to avoid generating many non-squares, digital root calc added. func recursehi(lst llst, lv int) {
lv1 := lv - 1
if lv == dl { // check if on last stage of permutation
sum = ac[lv1]
if (0x202021202030213 & (1 << (int(sum) & 63))) != 0 { // test accumulated sum, append to result if square
rt = int64(math.Sqrt(float64(sum)))
if rt*rt == sum {
sr = append(sr, sum)
}
}
} else {
for _, n := range lst[lv] { // set up next permutation
d[lv] = n
if lv == 0 {
ac[0] = pp[0] * int64(n)
dac[0] = drar[n] // update accumulated sum and running dr
} else {
ac[lv] = ac[lv1] + pp[lv]*int64(n)
dac[lv] = dac[lv1] + drar[n]
if dac[lv] > 8 {
dac[lv] -= 9
}
}
switch lv { // shortcuts to be performed on designated levels
case 0: // primary level: set shortcuts for secondary level
ln = n
lst[1] = lu[ln]
lst[2] = l2[n]
case 1: // secondary level: set shortcuts for tertiary level
switch ln { // for sums
case 5, 15:
if n < 10 {
lst[2] = evh
} else {
lst[2] = odh
}
case 9:
if ((n >> 1) & 1) == 0 {
lst[2] = evh
} else {
lst[2] = odh
}
case 11:
if ((n >> 1) & 1) == 1 {
lst[2] = evh
} else {
lst[2] = odh
}
}
}
if lv == dl-2 {
// reduce last round according to dr calc
if odd {
lst[dl-1] = chTen[dac[dl-2]]
} else {
lst[dl-1] = chAH[dac[dl-2]]
}
}
recursehi(lst, lv+1) // recursively call next level
}
}
}
// Recursive function to evaluate the lo permutations, shortcuts added to avoid // generating many non-squares. func recurselo(lst llst, lv int) {
lv1 := lv - 1
if lv == dl { // check if on last stage of permutation
sum = ac[lv1]
if sum > 0 {
rt = int64(math.Sqrt(float64(sum)))
if rt*rt == sum { // test accumulated sum, append to result if square
sr = append(sr, sum)
}
}
} else {
for _, n := range lst[lv] { // set up next permutation
d[lv] = n
if lv == 0 {
ac[0] = pp[0] * int64(n)
} else {
ac[lv] = ac[lv1] + pp[lv]*int64(n) // update accumulated sum
}
switch lv { // shortcuts to be performed on designated levels
case 0: // primary level: set shortcuts for secondary level
ln = n
lst[1] = lu[ln]
lst[2] = l2[n]
case 1: // secondary level: set shortcuts for tertiary level
switch ln { // for difs
case 1:
if (((n + 9) >> 1) & 1) == 0 {
lst[2] = evl
} else {
lst[2] = odl
}
case 5:
if n < 0 {
lst[2] = evl
} else {
lst[2] = odl
}
}
}
recurselo(lst, lv+1) // Recursively call next level
}
}
}
// Produces a list of candidate square numbers. func listEm(lst, plu, pl2 llst) []int64 {
dl = len(lst)
d = make([]int, dl)
sr = sr[:0]
lu = plu
l2 = pl2
ac = make([]int64, dl)
dac = make([]int, dl) // init support vars
pp = make([]int64, dl)
for i, j := 0, nd1; i < dl; i, j = i+1, j-1 {
// build coefficients array
if len(lst[0]) > 6 {
pp[i] = p[j] + p[i]
} else {
pp[i] = p[j] - p[i]
}
}
// call appropriate recursive function
if nd <= 5 {
recurseLE5(lst, 0)
} else if len(lst[0]) > 8 {
recursehi(lst, 0)
} else {
recurselo(lst, 0)
}
return sr
}
// Reveals whether combining two lists of squares can produce a Rare number. func reveal(lo, hi []int64) {
var s []string // create temp list of results
for _, l := range lo {
for _, h := range hi {
r := (h - l) >> 1
f := h - r // generate all possible fwd & rev candidates from lists
if isRev(nd, f, r) { // test and append sucesses to temp list
s = append(s, fmt.Sprintf("%20d %11d %10d ", f, isr(h), isr(l)))
}
}
}
sort.Strings(s)
if len(s) > 0 {
for _, t := range s { // if there are any, output sorted results
cn++
tt := ""
if t != s[len(s)-1] {
tt = "\n"
}
fmt.Printf("%2d %s%s", cn, t, tt)
}
} else {
fmt.Printf("%48s", "")
}
}
/* Unsigned variables and functions for nd == 19 */
var (
usum uint64 // unsigned calculated sum of terms (square candidate) urt uint64 // unsigned root of sum acu []uint64 // unsigned accumulator slice ppu []uint64 // unsigned long coefficient slice that combines with digits of working slice sru []uint64 // unsigned temporary list of squares used for building
)
// Returns Unsigned Integer Square Root. func isrU(s uint64) uint64 {
return uint64(math.Sqrt(float64(s)))
}
// Recursively determines whether 'r' is the reverse of 'f'. func isRevU(nd int, f, r uint64) bool {
nd--
if f/uint64(p[nd]) != r%10 {
return false
}
if nd < 1 {
return true
}
return isRevU(nd, f%uint64(p[nd]), r/10)
}
// Recursive function to evaluate the unsigned hi permutations, shortcuts added to avoid // generating many non-squares, digital root calc added. func recurseUhi(lst llst, lv int) {
lv1 := lv - 1
if lv == dl { // check if on last stage of permutation
usum = acu[lv1]
if (0x202021202030213 & (1 << (int(usum) & 63))) != 0 { // test accumulated sum, append to result if square
urt = uint64(math.Sqrt(float64(usum)))
if urt*urt == usum {
sru = append(sru, usum)
}
}
} else {
for _, n := range lst[lv] { // set up next permutation
d[lv] = n
if lv == 0 {
acu[0] = ppu[0] * uint64(n)
dac[0] = drar[n] // update accumulated sum and running dr
} else {
if n >= 0 {
acu[lv] = acu[lv1] + ppu[lv]*uint64(n)
} else {
acu[lv] = acu[lv1] - ppu[lv]*uint64(-n)
}
dac[lv] = dac[lv1] + drar[n]
if dac[lv] > 8 {
dac[lv] -= 9
}
}
switch lv { // shortcuts to be performed on designated levels
case 0: // primary level: set shortcuts for secondary level
ln = n
lst[1] = lu[ln]
lst[2] = l2[n]
case 1: // secondary level: set shortcuts for tertiary level
switch ln { // for sums
case 5, 15:
if n < 10 {
lst[2] = evh
} else {
lst[2] = odh
}
case 9:
if ((n >> 1) & 1) == 0 {
lst[2] = evh
} else {
lst[2] = odh
}
case 11:
if ((n >> 1) & 1) == 1 {
lst[2] = evh
} else {
lst[2] = odh
}
}
}
if lv == dl-2 {
// reduce last round according to dr calc
if odd {
lst[dl-1] = chTen[dac[dl-2]]
} else {
lst[dl-1] = chAH[dac[dl-2]]
}
}
recurseUhi(lst, lv+1) // recursively call next level
}
}
}
// Recursive function to evaluate the unsigned lo permutations, shortcuts added to avoid // generating many non-squares. func recurseUlo(lst llst, lv int) {
lv1 := lv - 1
if lv == dl { // check if on last stage of permutation
usum = acu[lv1]
if usum > 0 {
urt = uint64(math.Sqrt(float64(usum)))
if urt*urt == usum { // test accumulated sum, append to result if square
sru = append(sru, usum)
}
}
} else {
for _, n := range lst[lv] { // set up next permutation
d[lv] = n
if lv == 0 {
acu[0] = ppu[0] * uint64(n)
} else {
if n >= 0 {
acu[lv] = acu[lv1] + ppu[lv]*uint64(n) // update accumulated sum
} else {
acu[lv] = acu[lv1] - ppu[lv]*uint64(-n)
}
}
switch lv { // shortcuts to be performed on designated levels
case 0: // primary level: set shortcuts for secondary level
ln = n
lst[1] = lu[ln]
lst[2] = l2[n]
case 1: // secondary level: set shortcuts for tertiary level
switch ln { // for difs
case 1:
if (((n + 9) >> 1) & 1) == 0 {
lst[2] = evl
} else {
lst[2] = odl
}
case 5:
if n < 0 {
lst[2] = evl
} else {
lst[2] = odl
}
}
}
recurseUlo(lst, lv+1) // Recursively call next level
}
}
}
// Produces a list of candidate square numbers. func listEmU(lst, plu, pl2 llst) []uint64 {
dl = len(lst)
d = make([]int, dl)
sru = sru[:0]
lu = plu
l2 = pl2
acu = make([]uint64, dl)
dac = make([]int, dl) // init support vars
ppu = make([]uint64, dl)
for i, j := 0, nd1; i < dl; i, j = i+1, j-1 {
// build coefficients array
if len(lst[0]) > 6 {
ppu[i] = uint64(p[j] + p[i])
} else {
ppu[i] = uint64(p[j] - p[i])
}
}
// call appropriate recursive functin on
if len(lst[0]) > 8 {
recurseUhi(lst, 0)
} else {
recurseUlo(lst, 0)
}
return sru
}
// Reveals whether combining two lists of unsigned squares can produce a Rare number. func revealU(lo, hi []uint64) {
var s []string // create temp list of results
for _, l := range lo {
for _, h := range hi {
r := (h - l) >> 1
f := h - r // generate all possible fwd & rev candidates from lists
if isRevU(nd, f, r) { // test and append sucesses to temp list
s = append(s, fmt.Sprintf("%20d %11d %10d ", f, isrU(h), isrU(l)))
}
}
}
sort.Strings(s)
if len(s) > 0 {
for _, t := range s { // if there are any, output sorted results
cn++
tt := ""
if t != s[len(s)-1] {
tt = "\n"
}
fmt.Printf("%2d %s%s", cn, t, tt)
}
} else {
fmt.Printf("%48s", "")
}
}
var (
bStart time.Time // block start time tStart time.Time // total start time
)
// Formats time in form hh:mm:ss.fff (i.e. millisecond precision). func formatTime(d time.Duration) string {
f := d.Milliseconds()
s := f / 1000
f %= 1000
m := s / 60
s %= 60
h := m / 60
m %= 60
return fmt.Sprintf("%02d:%02d:%02d.%03d", h, m, s, f)
}
func main() {
start := time.Now()
fmt.Printf("%3s%20s %11s %10s %3s %11s %11s\n", "nth", "forward", "rt.sum", "rt.dif", "digs", "block time", "total time")
p[0] = 1
for i, j := 0, 1; j < len(p); j++ {
p[j] = p[i] * 10 // create powers of 10 array
i = j
}
for i := 0; i < len(drar); i++ {
drar[i] = (i << 1) % 9 // create digital root array
}
bStart = time.Now()
tStart = bStart
lls := llst{tlo}
hls := llst{thi}
for nd <= 18 { // loop through all numbers of digits
if nd > 2 {
if odd {
hls = append(hls, ten)
} else {
lls = append(lls, all)
hls[len(hls)-1] = alh
}
} // build permutations list
tmp1 := listEm(lls, lul, l2l)
tmp2 := make([]int64, len(tmp1))
copy(tmp2, tmp1)
reveal(tmp2, listEm(hls, luh, l2h)) // reveal results
if !odd && nd > 5 {
hls[len(hls)-1] = alh // restore last element of hls, so that dr shortcut doesn't mess up next nd
}
bTime := formatTime(time.Since(bStart))
tTime := formatTime(time.Since(tStart))
fmt.Printf("%2d: %s %s\n", nd, bTime, tTime)
bStart = time.Now() // restart block timing
nd1 = nd
nd++
odd = !odd
}
// nd == 19
hls = append(hls, ten)
tmp3 := listEmU(lls, lul, l2l)
tmp4 := make([]uint64, len(tmp3))
copy(tmp4, tmp3)
revealU(tmp4, listEmU(hls, luh, l2h)) // reveal unsigned results
fbTime := formatTime(time.Since(bStart))
ftTime := formatTime(time.Since(tStart))
fmt.Printf("%2d: %s %s\n", nd, fbTime, ftTime)
}</lang>
- Output:
nth forward rt.sum rt.dif digs block time total time
1 65 11 3 2: 00:00:00.000 00:00:00.000
3: 00:00:00.000 00:00:00.000
4: 00:00:00.000 00:00:00.000
5: 00:00:00.000 00:00:00.000
2 621770 836 738 6: 00:00:00.000 00:00:00.000
7: 00:00:00.000 00:00:00.000
8: 00:00:00.001 00:00:00.001
3 281089082 23708 330 9: 00:00:00.006 00:00:00.008
4 2022652202 63602 300
5 2042832002 63602 6360 10: 00:00:00.015 00:00:00.023
11: 00:00:00.036 00:00:00.060
6 868591084757 1275175 333333
7 872546974178 1320616 32670
8 872568754178 1320616 33330 12: 00:00:00.057 00:00:00.117
9 6979302951885 3586209 1047717 13: 00:00:00.376 00:00:00.494
10 20313693904202 6368252 269730
11 20313839704202 6368252 270270
12 20331657922202 6368252 329670
13 20331875722202 6368252 330330
14 20333875702202 6368252 336330
15 40313893704200 6368252 6330336
16 40351893720200 6368252 6336336 14: 00:00:00.981 00:00:01.475
17 200142385731002 20006998 69300
18 204238494066002 20122102 1891560
19 221462345754122 21045662 69300
20 244062891224042 22011022 1908060
21 245518996076442 22140228 921030
22 248359494187442 22206778 1891560
23 403058392434500 20211202 19940514
24 441054594034340 22011022 19940514
25 816984566129618 40421606 250800 15: 00:00:07.042 00:00:08.517
26 2078311262161202 64030648 7529850
27 2133786945766212 65272218 2666730
28 2135568943984212 65272218 3267330
29 2135764587964212 65272218 3326670
30 2135786765764212 65272218 3333330
31 4135786945764210 65272218 63333336
32 6157577986646405 105849161 33333333
33 6889765708183410 83866464 82133718
34 8052956026592517 123312255 29999997
35 8052956206592517 123312255 30000003
36 8191154686620818 127950856 3299670
37 8191156864620818 127950856 3300330
38 8191376864400818 127950856 3366330
39 8650327689541457 127246955 33299667
40 8650349867341457 127246955 33300333 16: 00:00:18.521 00:00:27.039
41 22542040692914522 212329862 333300
42 67725910561765640 269040196 251135808
43 86965750494756968 417050956 33000 17: 00:02:13.481 00:02:40.521
44 225342456863243522 671330638 297000
45 225342458663243522 671330638 303000
46 225342478643243522 671330638 363000
47 284684666566486482 754565658 30000
48 284684868364486482 754565658 636000
49 297128548234950692 770186978 32697330
50 297128722852950692 770186978 32702670
51 297148324656930692 770186978 33296670
52 297148546434930692 770186978 33303330
53 497168548234910690 770186978 633363336
54 619431353040136925 1071943279 299667003
55 619631153042134925 1071943279 300333003
56 631688638047992345 1083968809 297302703
57 633288858025996145 1083968809 302637303
58 633488632647994145 1083968809 303296697
59 653488856225994125 1083968809 363303363
60 811865096390477018 1273828556 33030330
61 865721270017296468 1315452006 32071170
62 871975098681469178 1320582934 3303300
63 898907259301737498 1339270086 64576740 18: 00:06:17.288 00:08:57.810
64 2042401829204402402 2021001202 18915600
65 2060303819041450202 2020110202 199405140
66 2420424089100600242 2200110022 19080600
67 2551755006254571552 2259094848 693000
68 2702373360882732072 2324811012 693000
69 2825378427312735282 2377130742 2508000
70 6531727101458000045 3454234451 1063822617
71 6988066446726832640 2729551744 2554541088
72 8066308349502036608 4016542096 2508000
73 8197906905009010818 4046976144 133408770
74 8200756128308135597 4019461925 495417087
75 8320411466598809138 4079154376 36366330 19: 00:45:42.006 00:54:39.816
Turbo
A smallish turbo anyway :)
The following, which is a translation of the C++ 10 to 19 digits program and includes improvements suggested by Enter your username (see Talk page), completes in around 15.25 minutes which is more than 3.5 times faster than the 'quicker' version above.
Curiously, the C++ program itself when compiled with g++ 7.5.0 (-std=c++17 -O3) on the same machine (and incorporating the same improvements) completes in just over 21 minutes though I understand that when using other C++ compilers (clang, mingw) it's much faster than this. <lang go>package main
import (
"fmt" "math" "sort" "time"
)
type (
z1 func() z2
z2 struct {
value int64
hasValue bool
}
)
var pow10 [19]int64
func init() {
pow10[0] = 1
for i := 1; i < 19; i++ {
pow10[i] = 10 * pow10[i-1]
}
}
func izRev(n int, i, g uint64) bool {
if i/uint64(pow10[n-1]) != g%10 {
return false
}
if n < 2 {
return true
}
return izRev(n-1, i%uint64(pow10[n-1]), g/10)
}
func fG(n z1, start, end, reset int, step int64, l *int64) z1 {
i, g, e := step*int64(start), step*int64(end), step*int64(reset)
return func() z2 {
for i < g {
*l += step
i += step
return z2{*l, true}
}
i = e
*l -= (g - e)
return n()
}
}
type nLH struct{ even, odd []uint64 }
type zp struct {
n z1 g [][2]int64
}
func newNLH(e zp) nLH {
var even, odd []uint64
n, g := e.n, e.g
for i := n(); i.hasValue; i = n() {
for _, p := range g {
ng, gg := p[0], p[1]
if (ng > 0) || (i.value > 0) {
w := uint64(ng*pow10[4] + gg + i.value)
ws := uint64(math.Sqrt(float64(w)))
if ws*ws == w {
if w%2 == 0 {
even = append(even, w)
} else {
odd = append(odd, w)
}
}
}
}
}
return nLH{even, odd}
}
func makeL(n int) zp {
g := make([]z1, n/2-3)
g[0] = func() z2 { return z2{} }
for i := 1; i < n/2-3; i++ {
s := -9
if i == n/2-4 {
s = -10
}
l := pow10[n-i-4] - pow10[i+3]
acc += l * int64(s)
g[i] = fG(g[i-1], s, 9, -9, l, &acc)
}
var g0, g1, g2, g3 int64
l0, l1, l2, l3 := pow10[n-5], pow10[n-6], pow10[n-7], pow10[n-8]
f := func() [][2]int64 {
var w [][2]int64
for g0 < 7 {
nn := g3*l3 + g2*l2 + g1*l1 + g0*l0
gg := -1000*g3 - 100*g2 - 10*g1 - g0
if g3 < 9 {
g3++
} else {
g3 = -9
if g2 < 9 {
g2++
} else {
g2 = -9
if g1 < 9 {
g1++
} else {
g1 = -9
if g0 == 1 {
g0 = 3
}
g0++
}
}
}
if bs[(pow10[10]+gg)%10000] {
w = append(w, [2]int64{nn, gg})
}
}
return w
}
return zp{g[n/2-4], f()}
}
func makeH(n int) zp {
acc = -(pow10[n/2] + pow10[(n-1)/2])
g := make([]z1, (n+1)/2-3)
g[0] = func() z2 { return z2{} }
for i := 1; i < n/2-3; i++ {
j := 0
if i == (n+1)/2-3 {
j = -1
}
g[i] = fG(g[i-1], j, 18, 0, pow10[n-i-4]+pow10[i+3], &acc)
if n%2 == 1 {
g[(n+1)/2-4] = fG(g[n/2-4], -1, 9, 0, 2*pow10[n/2], &acc)
}
}
g0 := int64(4)
var g1, g2, g3 int64
l0, l1, l2, l3 := pow10[n-5], pow10[n-6], pow10[n-7], pow10[n-8]
f := func() [][2]int64 {
var w [][2]int64
for g0 < 17 {
nn := g3*l3 + g2*l2 + g1*l1 + g0*l0
gg := 1000*g3 + 100*g2 + 10*g1 + g0
if g3 < 18 {
g3++
} else {
g3 = 0
if g2 < 18 {
g2++
} else {
g2 = 0
if g1 < 18 {
g1++
} else {
g1 = 0
if g0 == 6 || g0 == 9 {
g0 += 3
}
g0++
}
}
}
if bs[gg%10000] {
w = append(w, [2]int64{nn, gg})
}
}
return w
}
return zp{g[(n+1)/2-4], f()}
}
var (
acc int64 bs = make([]bool, 10000) L, H nLH
)
func rare(n int) []uint64 {
acc = 0
for g := 0; g < 10000; g++ {
bs[(g*g)%10000] = true
}
L = newNLH(makeL(n))
H = newNLH(makeH(n))
var rares []uint64
for _, l := range L.even {
for _, h := range H.even {
r := (h - l) / 2
z := h - r
if izRev(n, r, z) {
rares = append(rares, z)
}
}
}
for _, l := range L.odd {
for _, h := range H.odd {
r := (h - l) / 2
z := h - r
if izRev(n, r, z) {
rares = append(rares, z)
}
}
}
if len(rares) > 0 {
sort.Slice(rares, func(i, j int) bool {
return rares[i] < rares[j]
})
}
return rares
}
// Formats time in form hh:mm:ss.fff (i.e. millisecond precision). func formatTime(d time.Duration) string {
f := d.Milliseconds()
s := f / 1000
f %= 1000
m := s / 60
s %= 60
h := m / 60
m %= 60
return fmt.Sprintf("%02d:%02d:%02d.%03d", h, m, s, f)
}
func commatize(n uint64) string {
s := fmt.Sprintf("%d", n)
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
return s
}
func main() {
bStart := time.Now() // block time
tStart := bStart // total time
nth := 3 // i.e. count of rare numbers < 10 digits
fmt.Println("nth rare number digs block time total time")
for nd := 10; nd <= 19; nd++ {
rares := rare(nd)
if len(rares) > 0 {
for i, r := range rares {
nth++
t := ""
if i < len(rares)-1 {
t = "\n"
}
fmt.Printf("%2d %25s%s", nth, commatize(r), t)
}
} else {
fmt.Printf("%29s", "")
}
fbTime := formatTime(time.Since(bStart))
ftTime := formatTime(time.Since(tStart))
fmt.Printf(" %2d: %s %s\n", nd, fbTime, ftTime)
bStart = time.Now() // restart block timing
}
}</lang>
- Output:
nth rare number digs block time total time
4 2,022,652,202
5 2,042,832,002 10: 00:00:00.002 00:00:00.002
11: 00:00:00.008 00:00:00.011
6 868,591,084,757
7 872,546,974,178
8 872,568,754,178 12: 00:00:00.022 00:00:00.033
9 6,979,302,951,885 13: 00:00:00.097 00:00:00.131
10 20,313,693,904,202
11 20,313,839,704,202
12 20,331,657,922,202
13 20,331,875,722,202
14 20,333,875,702,202
15 40,313,893,704,200
16 40,351,893,720,200 14: 00:00:00.265 00:00:00.396
17 200,142,385,731,002
18 204,238,494,066,002
19 221,462,345,754,122
20 244,062,891,224,042
21 245,518,996,076,442
22 248,359,494,187,442
23 403,058,392,434,500
24 441,054,594,034,340
25 816,984,566,129,618 15: 00:00:01.774 00:00:02.170
26 2,078,311,262,161,202
27 2,133,786,945,766,212
28 2,135,568,943,984,212
29 2,135,764,587,964,212
30 2,135,786,765,764,212
31 4,135,786,945,764,210
32 6,157,577,986,646,405
33 6,889,765,708,183,410
34 8,052,956,026,592,517
35 8,052,956,206,592,517
36 8,191,154,686,620,818
37 8,191,156,864,620,818
38 8,191,376,864,400,818
39 8,650,327,689,541,457
40 8,650,349,867,341,457 16: 00:00:05.407 00:00:07.578
41 22,542,040,692,914,522
42 67,725,910,561,765,640
43 86,965,750,494,756,968 17: 00:00:35.401 00:00:42.979
44 225,342,456,863,243,522
45 225,342,458,663,243,522
46 225,342,478,643,243,522
47 284,684,666,566,486,482
48 284,684,868,364,486,482
49 297,128,548,234,950,692
50 297,128,722,852,950,692
51 297,148,324,656,930,692
52 297,148,546,434,930,692
53 497,168,548,234,910,690
54 619,431,353,040,136,925
55 619,631,153,042,134,925
56 631,688,638,047,992,345
57 633,288,858,025,996,145
58 633,488,632,647,994,145
59 653,488,856,225,994,125
60 811,865,096,390,477,018
61 865,721,270,017,296,468
62 871,975,098,681,469,178
63 898,907,259,301,737,498 18: 00:01:42.745 00:02:25.725
64 2,042,401,829,204,402,402
65 2,060,303,819,041,450,202
66 2,420,424,089,100,600,242
67 2,551,755,006,254,571,552
68 2,702,373,360,882,732,072
69 2,825,378,427,312,735,282
70 6,531,727,101,458,000,045
71 6,988,066,446,726,832,640
72 8,066,308,349,502,036,608
73 8,197,906,905,009,010,818
74 8,200,756,128,308,135,597
75 8,320,411,466,598,809,138 19: 00:12:48.590 00:15:14.316
J
The following function determines whether a given integer is "rare", as can be seen in the sample output.
To use it to find rare numbers, one would simply apply it to each integer seriatim, and keep the numbers where its output is true (I.@:rare i. AS_HIGH_AS_YOU_WANT_TO_CHECK); but since these numbers are, well, rare, actually doing so would take a very long time.
<lang j>rare =: ( np@:] *. (nbrPs rr) ) b10
np =: -.@:(-: |.) NB. Not palindromic nbrPs =: > *. sdPs NB. n is Bigger than R and the perfect square constraint is satisfied sdPs =: + *.&:ps - NB. n > rr and both their sum and difference are perfect squares ps =: 0 = 1 | %: NB. Perfect square (integral sqrt) rr =: 10&#.@:|. NB. Do note we do reverse the digits twice (once here, once in np) b10 =: 10&#.^:_1 NB. Base 10 digits</lang>
- Output:
<lang j> NB. From OEIS
R =: 65 621770 281089082 2022652202 2042832002 868591084757 872546974178 872568754178 6979302951885 20313693904202 20313839704202 20331657922202 20331875722202 20333875702202 40313893704200 rare"0 R
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1</lang>
Java
<lang java>import java.time.Duration; import java.time.LocalDateTime; import java.util.ArrayList; import java.util.Collections; import java.util.HashMap; import java.util.List; import java.util.Map; import java.util.concurrent.atomic.AtomicInteger; import java.util.concurrent.atomic.AtomicReference;
public class RareNumbers {
public interface Consumer5<A, B, C, D, E> {
void apply(A a, B b, C c, D d, E e);
}
public interface Consumer7<A, B, C, D, E, F, G> {
void apply(A a, B b, C c, D d, E e, F f, G g);
}
public interface Recursable5<A, B, C, D, E> {
void apply(A a, B b, C c, D d, E e, Recursable5<A, B, C, D, E> r);
}
public interface Recursable7<A, B, C, D, E, F, G> {
void apply(A a, B b, C c, D d, E e, F f, G g, Recursable7<A, B, C, D, E, F, G> r);
}
public static <A, B, C, D, E> Consumer5<A, B, C, D, E> recurse(Recursable5<A, B, C, D, E> r) {
return (a, b, c, d, e) -> r.apply(a, b, c, d, e, r);
}
public static <A, B, C, D, E, F, G> Consumer7<A, B, C, D, E, F, G> recurse(Recursable7<A, B, C, D, E, F, G> r) {
return (a, b, c, d, e, f, g) -> r.apply(a, b, c, d, e, f, g, r);
}
private static class Term {
long coeff;
byte ix1, ix2;
public Term(long coeff, byte ix1, byte ix2) {
this.coeff = coeff;
this.ix1 = ix1;
this.ix2 = ix2;
}
}
private static final int MAX_DIGITS = 16;
private static long toLong(List<Byte> digits, boolean reverse) {
long sum = 0;
if (reverse) {
for (int i = digits.size() - 1; i >= 0; --i) {
sum = sum * 10 + digits.get(i);
}
} else {
for (Byte digit : digits) {
sum = sum * 10 + digit;
}
}
return sum;
}
private static boolean isNotSquare(long n) {
long root = (long) Math.sqrt(n);
return root * root != n;
}
private static List<Byte> seq(byte from, byte to, byte step) {
List<Byte> res = new ArrayList<>();
for (byte i = from; i <= to; i += step) {
res.add(i);
}
return res;
}
private static String commatize(long n) {
String s = String.valueOf(n);
int le = s.length();
int i = le - 3;
while (i >= 1) {
s = s.substring(0, i) + "," + s.substring(i);
i -= 3;
}
return s;
}
public static void main(String[] args) {
final LocalDateTime startTime = LocalDateTime.now();
long pow = 1L;
System.out.println("Aggregate timings to process all numbers up to:");
// terms of (n-r) expression for number of digits from 2 to maxDigits
List<List<Term>> allTerms = new ArrayList<>();
for (int i = 0; i < MAX_DIGITS - 1; ++i) {
allTerms.add(new ArrayList<>());
}
for (int r = 2; r <= MAX_DIGITS; ++r) {
List<Term> terms = new ArrayList<>();
pow *= 10;
long pow1 = pow;
long pow2 = 1;
byte i1 = 0;
byte i2 = (byte) (r - 1);
while (i1 < i2) {
terms.add(new Term(pow1 - pow2, i1, i2));
pow1 /= 10;
pow2 *= 10;
i1++;
i2--;
}
allTerms.set(r - 2, terms);
}
// map of first minus last digits for 'n' to pairs giving this value
Map<Byte, List<List<Byte>>> fml = Map.of(
(byte) 0, List.of(List.of((byte) 2, (byte) 2), List.of((byte) 8, (byte) 8)),
(byte) 1, List.of(List.of((byte) 6, (byte) 5), List.of((byte) 8, (byte) 7)),
(byte) 4, List.of(List.of((byte) 4, (byte) 0)),
(byte) 6, List.of(List.of((byte) 6, (byte) 0), List.of((byte) 8, (byte) 2))
);
// map of other digit differences for 'n' to pairs giving this value
Map<Byte, List<List<Byte>>> dmd = new HashMap<>();
for (int i = 0; i < 100; ++i) {
List<Byte> a = List.of((byte) (i / 10), (byte) (i % 10));
int d = a.get(0) - a.get(1);
dmd.computeIfAbsent((byte) d, k -> new ArrayList<>()).add(a);
}
List<Byte> fl = List.of((byte) 0, (byte) 1, (byte) 4, (byte) 6);
List<Byte> dl = seq((byte) -9, (byte) 9, (byte) 1); // all differences
List<Byte> zl = List.of((byte) 0); // zero differences only
List<Byte> el = seq((byte) -8, (byte) 8, (byte) 2); // even differences only
List<Byte> ol = seq((byte) -9, (byte) 9, (byte) 2); // odd differences only
List<Byte> il = seq((byte) 0, (byte) 9, (byte) 1);
List<Long> rares = new ArrayList<>();
List<List<List<Byte>>> lists = new ArrayList<>();
for (int i = 0; i < 4; ++i) {
lists.add(new ArrayList<>());
}
for (int i = 0; i < fl.size(); ++i) {
List<List<Byte>> temp1 = new ArrayList<>();
List<Byte> temp2 = new ArrayList<>();
temp2.add(fl.get(i));
temp1.add(temp2);
lists.set(i, temp1);
}
final AtomicReference<List<Byte>> digits = new AtomicReference<>(new ArrayList<>());
AtomicInteger count = new AtomicInteger();
// Recursive closure to generate (n+r) candidates from (n-r) candidates
// and hence find Rare numbers with a given number of digits.
Consumer7<List<Byte>, List<Byte>, List<List<Byte>>, List<List<Byte>>, Long, Integer, Integer> fnpr = recurse((cand, di, dis, indicies, nmr, nd, level, func) -> {
if (level == dis.size()) {
digits.get().set(indicies.get(0).get(0), fml.get(cand.get(0)).get(di.get(0)).get(0));
digits.get().set(indicies.get(0).get(1), fml.get(cand.get(0)).get(di.get(0)).get(1));
int le = di.size();
if (nd % 2 == 1) {
le--;
digits.get().set(nd / 2, di.get(le));
}
for (int i = 1; i < le; ++i) {
digits.get().set(indicies.get(i).get(0), dmd.get(cand.get(i)).get(di.get(i)).get(0));
digits.get().set(indicies.get(i).get(1), dmd.get(cand.get(i)).get(di.get(i)).get(1));
}
long r = toLong(digits.get(), true);
long npr = nmr + 2 * r;
if (isNotSquare(npr)) {
return;
}
count.getAndIncrement();
System.out.printf(" R/N %2d:", count.get());
LocalDateTime checkPoint = LocalDateTime.now();
long elapsed = Duration.between(startTime, checkPoint).toMillis();
System.out.printf(" %9sms", elapsed);
long n = toLong(digits.get(), false);
System.out.printf(" (%s)\n", commatize(n));
rares.add(n);
} else {
for (Byte num : dis.get(level)) {
di.set(level, num);
func.apply(cand, di, dis, indicies, nmr, nd, level + 1, func);
}
}
});
// Recursive closure to generate (n-r) candidates with a given number of digits.
Consumer5<List<Byte>, List<List<Byte>>, List<List<Byte>>, Integer, Integer> fnmr = recurse((cand, list, indicies, nd, level, func) -> {
if (level == list.size()) {
long nmr = 0;
long nmr2 = 0;
List<Term> terms = allTerms.get(nd - 2);
for (int i = 0; i < terms.size(); ++i) {
Term t = terms.get(i);
if (cand.get(i) >= 0) {
nmr += t.coeff * cand.get(i);
} else {
nmr2 += t.coeff * -cand.get(i);
if (nmr >= nmr2) {
nmr -= nmr2;
nmr2 = 0;
} else {
nmr2 -= nmr;
nmr = 0;
}
}
}
if (nmr2 >= nmr) {
return;
}
nmr -= nmr2;
if (isNotSquare(nmr)) {
return;
}
List<List<Byte>> dis = new ArrayList<>();
dis.add(seq((byte) 0, (byte) (fml.get(cand.get(0)).size() - 1), (byte) 1));
for (int i = 1; i < cand.size(); ++i) {
dis.add(seq((byte) 0, (byte) (dmd.get(cand.get(i)).size() - 1), (byte) 1));
}
if (nd % 2 == 1) {
dis.add(il);
}
List<Byte> di = new ArrayList<>();
for (int i = 0; i < dis.size(); ++i) {
di.add((byte) 0);
}
fnpr.apply(cand, di, dis, indicies, nmr, nd, 0);
} else {
for (Byte num : list.get(level)) {
cand.set(level, num);
func.apply(cand, list, indicies, nd, level + 1, func);
}
}
});
for (int nd = 2; nd <= MAX_DIGITS; ++nd) {
digits.set(new ArrayList<>());
for (int i = 0; i < nd; ++i) {
digits.get().add((byte) 0);
}
if (nd == 4) {
lists.get(0).add(zl);
lists.get(1).add(ol);
lists.get(2).add(el);
lists.get(3).add(ol);
} else if (allTerms.get(nd - 2).size() > lists.get(0).size()) {
for (int i = 0; i < 4; ++i) {
lists.get(i).add(dl);
}
}
List<List<Byte>> indicies = new ArrayList<>();
for (Term t : allTerms.get(nd - 2)) {
indicies.add(List.of(t.ix1, t.ix2));
}
for (List<List<Byte>> list : lists) {
List<Byte> cand = new ArrayList<>();
for (int i = 0; i < list.size(); ++i) {
cand.add((byte) 0);
}
fnmr.apply(cand, list, indicies, nd, 0);
}
LocalDateTime checkPoint = LocalDateTime.now();
long elapsed = Duration.between(startTime, checkPoint).toMillis();
System.out.printf(" %2d digits: %9sms\n", nd, elapsed);
}
Collections.sort(rares);
System.out.printf("\nThe rare numbers with up to %d digits are:\n", MAX_DIGITS);
for (int i = 0; i < rares.size(); ++i) {
System.out.printf(" %2d: %25s\n", i + 1, commatize(rares.get(i)));
}
}
}</lang>
- Output:
Aggregate timings to process all numbers up to:
R/N 1: 17ms (65)
2 digits: 19ms
3 digits: 20ms
4 digits: 21ms
5 digits: 21ms
R/N 2: 29ms (621,770)
6 digits: 55ms
7 digits: 57ms
8 digits: 72ms
R/N 3: 76ms (281,089,082)
9 digits: 85ms
R/N 4: 89ms (2,022,652,202)
R/N 5: 237ms (2,042,832,002)
10 digits: 451ms
11 digits: 503ms
R/N 6: 833ms (872,546,974,178)
R/N 7: 869ms (872,568,754,178)
R/N 8: 1324ms (868,591,084,757)
12 digits: 1582ms
R/N 9: 1888ms (6,979,302,951,885)
13 digits: 2299ms
R/N 10: 6199ms (20,313,693,904,202)
R/N 11: 6272ms (20,313,839,704,202)
R/N 12: 7831ms (20,331,657,922,202)
R/N 13: 8070ms (20,331,875,722,202)
R/N 14: 8708ms (20,333,875,702,202)
R/N 15: 19681ms (40,313,893,704,200)
R/N 16: 19823ms (40,351,893,720,200)
14 digits: 21712ms
R/N 17: 21758ms (200,142,385,731,002)
R/N 18: 21991ms (221,462,345,754,122)
R/N 19: 24990ms (816,984,566,129,618)
R/N 20: 26552ms (245,518,996,076,442)
R/N 21: 26774ms (204,238,494,066,002)
R/N 22: 26846ms (248,359,494,187,442)
R/N 23: 27158ms (244,062,891,224,042)
R/N 24: 32349ms (403,058,392,434,500)
R/N 25: 32576ms (441,054,594,034,340)
15 digits: 34465ms
R/N 26: 85843ms (2,133,786,945,766,212)
R/N 27: 105944ms (2,135,568,943,984,212)
R/N 28: 109027ms (8,191,154,686,620,818)
R/N 29: 111677ms (8,191,156,864,620,818)
R/N 30: 112849ms (2,135,764,587,964,212)
R/N 31: 114572ms (2,135,786,765,764,212)
R/N 32: 118622ms (8,191,376,864,400,818)
R/N 33: 132237ms (2,078,311,262,161,202)
R/N 34: 164708ms (8,052,956,026,592,517)
R/N 35: 169421ms (8,052,956,206,592,517)
R/N 36: 203280ms (8,650,327,689,541,457)
R/N 37: 205555ms (8,650,349,867,341,457)
R/N 38: 207237ms (6,157,577,986,646,405)
R/N 39: 246082ms (4,135,786,945,764,210)
R/N 40: 275691ms (6,889,765,708,183,410)
16 digits: 278088ms
The rare numbers with up to 16 digits are:
1: 65
2: 621,770
3: 281,089,082
4: 2,022,652,202
5: 2,042,832,002
6: 868,591,084,757
7: 872,546,974,178
8: 872,568,754,178
9: 6,979,302,951,885
10: 20,313,693,904,202
11: 20,313,839,704,202
12: 20,331,657,922,202
13: 20,331,875,722,202
14: 20,333,875,702,202
15: 40,313,893,704,200
16: 40,351,893,720,200
17: 200,142,385,731,002
18: 204,238,494,066,002
19: 221,462,345,754,122
20: 244,062,891,224,042
21: 245,518,996,076,442
22: 248,359,494,187,442
23: 403,058,392,434,500
24: 441,054,594,034,340
25: 816,984,566,129,618
26: 2,078,311,262,161,202
27: 2,133,786,945,766,212
28: 2,135,568,943,984,212
29: 2,135,764,587,964,212
30: 2,135,786,765,764,212
31: 4,135,786,945,764,210
32: 6,157,577,986,646,405
33: 6,889,765,708,183,410
34: 8,052,956,026,592,517
35: 8,052,956,206,592,517
36: 8,191,154,686,620,818
37: 8,191,156,864,620,818
38: 8,191,376,864,400,818
39: 8,650,327,689,541,457
40: 8,650,349,867,341,457
Julia
<lang julia>using Formatting, Printf
struct Term
coeff::UInt64 ix1::Int8 ix2::Int8
end
function toUInt64(dgits, reverse)
return reverse ? foldr((i, j) -> i + 10j, UInt64.(dgits)) :
foldl((i, j) -> 10i + j, UInt64.(dgits))
end
function issquare(n)
if 0x202021202030213 & (1 << (UInt64(n) & 63)) != 0
root = UInt64(floor(sqrt(n)))
return root * root == n
end
return false
end
seq(from, to, step) = Int8.(collect(from:step:to))
commatize(n::Integer) = format(n, commas=true)
const verbose = true const count = [0]
""" Recursive closure to generate (n+r) candidates from (n-r) candidates and hence find Rare numbers with a given number of digits. """ function fnpr(cand, di, dis, indices, nmr, nd, level, dgits, fml, dmd, start, rares, il)
if level == length(dis)
dgits[indices[1][1] + 1] = fml[cand[1]][di[1] + 1][1]
dgits[indices[1][2] + 1] = fml[cand[1]][di[1] + 1][2]
le = length(di)
if nd % 2 == 1
le -= 1
dgits[nd ÷ 2 + 1] = di[le + 1]
end
for (i, d) in enumerate(di[2:le])
dgits[indices[i+1][1] + 1] = dmd[cand[i+1]][d + 1][1]
dgits[indices[i+1][2] + 1] = dmd[cand[i+1]][d + 1][2]
end
r = toUInt64(dgits, true)
npr = nmr + 2 * r
!issquare(npr) && return
count[1] += 1
verbose && @printf(" R/N %2d:", count[1])
!verbose && print("$count rares\b\b\b\b\b\b\b\b\b")
ms = UInt64(time() * 1000 - start)
verbose && @printf(" %9s ms", commatize(Int(ms)))
n = toUInt64(dgits, false)
verbose && @printf(" (%s)\n", commatize(BigInt(n)))
push!(rares, n)
else
for num in dis[level + 1]
di[level + 1] = num
fnpr(cand, di, dis, indices, nmr, nd, level + 1, dgits, fml, dmd, start, rares, il)
end
end
end # function fnpr
- Recursive closure to generate (n-r) candidates with a given number of digits.
- var fnmr func(cand []int8, list [][]int8, indices [][2]int8, nd, level int)
function fnmr(cand, list, indices, nd, level, allterms, fml, dmd, dgits, start, rares, il)
if level == length(list)
nmr, nmr2 = zero(UInt64), zero(UInt64)
for (i, t) in enumerate(allterms[nd - 1])
if cand[i] >= 0
nmr += t.coeff * UInt64(cand[i])
else
nmr2 += t.coeff * UInt64(-cand[i])
if nmr >= nmr2
nmr -= nmr2
nmr2 = zero(nmr2)
else
nmr2 -= nmr
nmr = zero(nmr)
end
end
end
nmr2 >= nmr && return
nmr -= nmr2
!issquare(nmr) && return
dis = [[seq(0, Int8(length(fml[cand[1]]) - 1), 1)] ;
[seq(0, Int8(length(dmd[c]) - 1), 1) for c in cand[2:end]]]
isodd(nd) && push!(dis, il)
di = zeros(Int8, length(dis))
fnpr(cand, di, dis, indices, nmr, nd, 0, dgits, fml, dmd, start, rares, il)
else
for num in list[level + 1]
cand[level + 1] = num
fnmr(cand, list, indices, nd, level + 1, allterms, fml, dmd, dgits, start, rares, il)
end
end
end # function fnmr
function findrare(maxdigits = 19)
start = time() * 1000.0
pow = one(UInt64)
verbose && println("Aggregate timings to process all numbers up to:")
# terms of (n-r) expression for number of digits from 2 to maxdigits
allterms = Vector{Vector{Term}}()
for r in 2:maxdigits
terms = Term[]
pow *= 10
pow1, pow2, i1, i2 = pow, one(UInt64), zero(Int8), Int8(r - 1)
while i1 < i2
push!(terms, Term(pow1 - pow2, i1, i2))
pow1, pow2, i1, i2 = pow1 ÷ 10, pow2 * 10, i1 + 1, i2 - 1
end
push!(allterms, terms)
end
# map of first minus last digits for 'n' to pairs giving this value
fml = Dict(
0 => [2 => 2, 8 => 8],
1 => [6 => 5, 8 => 7],
4 => [4 => 0],
6 => [6 => 0, 8 => 2],
)
# map of other digit differences for 'n' to pairs giving this value
dmd = Dict{Int8, Vector{Vector{Int8}}}()
for i in 0:99
a = [Int8(i ÷ 10), Int8(i % 10)]
d = a[1] - a[2]
v = get!(dmd, d, [])
push!(v, a)
end
fl = Int8[0, 1, 4, 6]
dl = seq(-9, 9, 1) # all differences
zl = Int8[0] # zero differences only
el = seq(-8, 8, 2) # even differences only
ol = seq(-9, 9, 2) # odd differences only
il = seq(0, 9, 1)
rares = UInt64[]
lists = [[[f]] for f in fl]
dgits = Int8[]
count[1] = 0
for nd = 2:maxdigits
dgits = zeros(Int8, nd)
if nd == 4
push!(lists[1], zl)
push!(lists[2], ol)
push!(lists[3], el)
push!(lists[4], ol)
elseif length(allterms[nd - 1]) > length(lists[1])
for i in 1:4
push!(lists[i], dl)
end
end
indices = Vector{Vector{Int8}}()
for t in allterms[nd - 1]
push!(indices, Int8[t.ix1, t.ix2])
end
for list in lists
cand = zeros(Int8, length(list))
fnmr(cand, list, indices, nd, 0, allterms, fml, dmd, dgits, start, rares, il)
end
ms = UInt64(time() * 1000 - start)
verbose && @printf(" %2d digits: %9s ms\n", nd, commatize(Int(ms)))
end
sort!(rares)
@printf("\nThe rare numbers with up to %d digits are:\n", maxdigits)
for (i, rare) in enumerate(rares)
@printf(" %2d: %25s\n", i, commatize(BigInt(rare)))
end
end # findrare function
findrare()
</lang>
- Output:
Timings are on a 2.9 GHz i5 processor, 16 Gb RAM, under Windows 10.
Aggregate timings to process all numbers up to:
R/N 1: 5 ms (65)
2 digits: 132 ms
3 digits: 133 ms
4 digits: 134 ms
5 digits: 134 ms
R/N 2: 135 ms (621,770)
6 digits: 135 ms
7 digits: 136 ms
8 digits: 140 ms
R/N 3: 141 ms (281,089,082)
9 digits: 143 ms
R/N 4: 144 ms (2,022,652,202)
R/N 5: 168 ms (2,042,832,002)
10 digits: 209 ms
11 digits: 251 ms
R/N 6: 443 ms (872,546,974,178)
R/N 7: 467 ms (872,568,754,178)
R/N 8: 773 ms (868,591,084,757)
12 digits: 919 ms
R/N 9: 1,178 ms (6,979,302,951,885)
13 digits: 1,510 ms
R/N 10: 4,662 ms (20,313,693,904,202)
R/N 11: 4,722 ms (20,313,839,704,202)
R/N 12: 6,028 ms (20,331,657,922,202)
R/N 13: 6,223 ms (20,331,875,722,202)
R/N 14: 6,753 ms (20,333,875,702,202)
R/N 15: 15,475 ms (40,313,893,704,200)
R/N 16: 15,594 ms (40,351,893,720,200)
14 digits: 16,749 ms
R/N 17: 16,772 ms (200,142,385,731,002)
R/N 18: 17,006 ms (221,462,345,754,122)
R/N 19: 20,027 ms (816,984,566,129,618)
R/N 20: 21,669 ms (245,518,996,076,442)
R/N 21: 21,895 ms (204,238,494,066,002)
R/N 22: 21,974 ms (248,359,494,187,442)
R/N 23: 22,302 ms (244,062,891,224,042)
R/N 24: 27,158 ms (403,058,392,434,500)
R/N 25: 27,405 ms (441,054,594,034,340)
15 digits: 28,744 ms
R/N 26: 79,350 ms (2,133,786,945,766,212)
R/N 27: 99,360 ms (2,135,568,943,984,212)
R/N 28: 102,426 ms (8,191,154,686,620,818)
R/N 29: 105,135 ms (8,191,156,864,620,818)
R/N 30: 106,334 ms (2,135,764,587,964,212)
R/N 31: 108,038 ms (2,135,786,765,764,212)
R/N 32: 112,142 ms (8,191,376,864,400,818)
R/N 33: 125,607 ms (2,078,311,262,161,202)
R/N 34: 154,417 ms (8,052,956,026,592,517)
R/N 35: 159,075 ms (8,052,956,206,592,517)
R/N 36: 192,323 ms (8,650,327,689,541,457)
R/N 37: 194,651 ms (8,650,349,867,341,457)
R/N 38: 196,344 ms (6,157,577,986,646,405)
R/N 39: 227,492 ms (4,135,786,945,764,210)
R/N 40: 244,502 ms (6,889,765,708,183,410)
16 digits: 245,658 ms
R/N 41: 251,178 ms (86,965,750,494,756,968)
R/N 42: 252,157 ms (22,542,040,692,914,522)
R/N 43: 382,883 ms (67,725,910,561,765,640)
17 digits: 393,371 ms
R/N 44: 427,555 ms (284,684,666,566,486,482)
R/N 45: 549,740 ms (225,342,456,863,243,522)
R/N 46: 594,392 ms (225,342,458,663,243,522)
R/N 47: 688,221 ms (225,342,478,643,243,522)
R/N 48: 753,385 ms (284,684,868,364,486,482)
R/N 49: 1,108,538 ms (871,975,098,681,469,178)
R/N 50: 1,770,255 ms (865,721,270,017,296,468)
R/N 51: 1,785,243 ms (297,128,548,234,950,692)
R/N 52: 1,793,571 ms (297,128,722,852,950,692)
R/N 53: 1,892,872 ms (811,865,096,390,477,018)
R/N 54: 1,941,208 ms (297,148,324,656,930,692)
R/N 55: 1,964,502 ms (297,148,546,434,930,692)
R/N 56: 2,267,616 ms (898,907,259,301,737,498)
R/N 57: 2,677,207 ms (631,688,638,047,992,345)
R/N 58: 2,702,836 ms (619,431,353,040,136,925)
R/N 59: 2,960,274 ms (619,631,153,042,134,925)
R/N 60: 3,019,846 ms (633,288,858,025,996,145)
R/N 61: 3,084,695 ms (633,488,632,647,994,145)
R/N 62: 3,924,801 ms (653,488,856,225,994,125)
R/N 63: 4,229,162 ms (497,168,548,234,910,690)
18 digits: 4,563,375 ms
R/N 64: 4,643,118 ms (2,551,755,006,254,571,552)
R/N 65: 4,662,645 ms (2,702,373,360,882,732,072)
R/N 66: 4,925,324 ms (2,825,378,427,312,735,282)
R/N 67: 4,947,368 ms (8,066,308,349,502,036,608)
R/N 68: 5,170,716 ms (2,042,401,829,204,402,402)
R/N 69: 5,216,832 ms (2,420,424,089,100,600,242)
R/N 70: 5,329,680 ms (8,320,411,466,598,809,138)
R/N 71: 5,634,991 ms (8,197,906,905,009,010,818)
R/N 72: 5,665,799 ms (2,060,303,819,041,450,202)
R/N 73: 5,861,019 ms (8,200,756,128,308,135,597)
R/N 74: 6,136,091 ms (6,531,727,101,458,000,045)
R/N 75: 6,770,242 ms (6,988,066,446,726,832,640)
19 digits: 6,846,705 ms
The rare numbers with up to 19 digits are:
1: 65
2: 621,770
3: 281,089,082
4: 2,022,652,202
5: 2,042,832,002
6: 868,591,084,757
7: 872,546,974,178
8: 872,568,754,178
9: 6,979,302,951,885
10: 20,313,693,904,202
11: 20,313,839,704,202
12: 20,331,657,922,202
13: 20,331,875,722,202
14: 20,333,875,702,202
15: 40,313,893,704,200
16: 40,351,893,720,200
17: 200,142,385,731,002
18: 204,238,494,066,002
19: 221,462,345,754,122
20: 244,062,891,224,042
21: 245,518,996,076,442
22: 248,359,494,187,442
23: 403,058,392,434,500
24: 441,054,594,034,340
25: 816,984,566,129,618
26: 2,078,311,262,161,202
27: 2,133,786,945,766,212
28: 2,135,568,943,984,212
29: 2,135,764,587,964,212
30: 2,135,786,765,764,212
31: 4,135,786,945,764,210
32: 6,157,577,986,646,405
33: 6,889,765,708,183,410
34: 8,052,956,026,592,517
35: 8,052,956,206,592,517
36: 8,191,154,686,620,818
37: 8,191,156,864,620,818
38: 8,191,376,864,400,818
39: 8,650,327,689,541,457
40: 8,650,349,867,341,457
41: 22,542,040,692,914,522
42: 67,725,910,561,765,640
43: 86,965,750,494,756,968
44: 225,342,456,863,243,522
45: 225,342,458,663,243,522
46: 225,342,478,643,243,522
47: 284,684,666,566,486,482
48: 284,684,868,364,486,482
49: 297,128,548,234,950,692
50: 297,128,722,852,950,692
51: 297,148,324,656,930,692
52: 297,148,546,434,930,692
53: 497,168,548,234,910,690
54: 619,431,353,040,136,925
55: 619,631,153,042,134,925
56: 631,688,638,047,992,345
57: 633,288,858,025,996,145
58: 633,488,632,647,994,145
59: 653,488,856,225,994,125
60: 811,865,096,390,477,018
61: 865,721,270,017,296,468
62: 871,975,098,681,469,178
63: 898,907,259,301,737,498
64: 2,042,401,829,204,402,402
65: 2,060,303,819,041,450,202
66: 2,420,424,089,100,600,242
67: 2,551,755,006,254,571,552
68: 2,702,373,360,882,732,072
69: 2,825,378,427,312,735,282
70: 6,531,727,101,458,000,045
71: 6,988,066,446,726,832,640
72: 8,066,308,349,502,036,608
73: 8,197,906,905,009,010,818
74: 8,200,756,128,308,135,597
75: 8,320,411,466,598,809,138
Kotlin
<lang scala>import java.time.Duration import java.time.LocalDateTime import kotlin.math.sqrt
class Term(var coeff: Long, var ix1: Byte, var ix2: Byte)
const val maxDigits = 16
fun toLong(digits: List<Byte>, reverse: Boolean): Long {
var sum: Long = 0
if (reverse) {
var i = digits.size - 1
while (i >= 0) {
sum = sum * 10 + digits[i]
i--
}
} else {
var i = 0
while (i < digits.size) {
sum = sum * 10 + digits[i]
i++
}
}
return sum
}
fun isSquare(n: Long): Boolean {
val root = sqrt(n.toDouble()).toLong() return root * root == n
}
fun seq(from: Byte, to: Byte, step: Byte): List<Byte> {
val res = mutableListOf<Byte>()
var i = from
while (i <= to) {
res.add(i)
i = (i + step).toByte()
}
return res
}
fun commatize(n: Long): String {
var s = n.toString()
val le = s.length
var i = le - 3
while (i >= 1) {
s = s.slice(0 until i) + "," + s.substring(i)
i -= 3
}
return s
}
fun main() {
val startTime = LocalDateTime.now()
var pow = 1L
println("Aggregate timings to process all numbers up to:")
// terms of (n-r) expression for number of digits from 2 to maxDigits
val allTerms = mutableListOf<MutableList<Term>>()
for (i in 0 until maxDigits - 1) {
allTerms.add(mutableListOf())
}
for (r in 2..maxDigits) {
val terms = mutableListOf<Term>()
pow *= 10
var pow1 = pow
var pow2 = 1L
var i1: Byte = 0
var i2 = (r - 1).toByte()
while (i1 < i2) {
terms.add(Term(pow1 - pow2, i1, i2))
pow1 /= 10
pow2 *= 10
i1++
i2--
}
allTerms[r - 2] = terms
}
// map of first minus last digits for 'n' to pairs giving this value
val fml = mapOf(
0.toByte() to listOf(listOf<Byte>(2, 2), listOf<Byte>(8, 8)),
1.toByte() to listOf(listOf<Byte>(6, 5), listOf<Byte>(8, 7)),
4.toByte() to listOf(listOf<Byte>(4, 0)),
6.toByte() to listOf(listOf<Byte>(6, 0), listOf<Byte>(8, 2))
)
// map of other digit differences for 'n' to pairs giving this value
val dmd = mutableMapOf<Byte, MutableList<List<Byte>>>()
for (i in 0 until 100) {
val a = listOf((i / 10).toByte(), (i % 10).toByte())
val d = a[0] - a[1]
dmd.getOrPut(d.toByte(), { mutableListOf() }).add(a)
}
val fl = listOf<Byte>(0, 1, 4, 6)
val dl = seq(-9, 9, 1) // all differences
val zl = listOf<Byte>(0) // zero differences only
val el = seq(-8, 8, 2) // even differences only
val ol = seq(-9, 9, 2) // odd differences only
val il = seq(0, 9, 1)
val rares = mutableListOf<Long>()
val lists = mutableListOf<MutableList<List<Byte>>>()
for (i in 0 until 4) {
lists.add(mutableListOf())
}
for (i_f in fl.withIndex()) {
lists[i_f.index] = mutableListOf(listOf(i_f.value))
}
var digits = mutableListOf<Byte>()
var count = 0
// Recursive closure to generate (n+r) candidates from (n-r) candidates
// and hence find Rare numbers with a given number of digits.
fun fnpr(
cand: List<Byte>,
di: MutableList<Byte>,
dis: List<List<Byte>>,
indicies: List<List<Byte>>,
nmr: Long,
nd: Int,
level: Int
) {
if (level == dis.size) {
digits[indicies[0][0].toInt()] = fml[cand[0]]?.get(di[0].toInt())?.get(0)!!
digits[indicies[0][1].toInt()] = fml[cand[0]]?.get(di[0].toInt())?.get(1)!!
var le = di.size
if (nd % 2 == 1) {
le--
digits[nd / 2] = di[le]
}
for (i_d in di.slice(1 until le).withIndex()) {
digits[indicies[i_d.index + 1][0].toInt()] = dmd[cand[i_d.index + 1]]?.get(i_d.value.toInt())?.get(0)!!
digits[indicies[i_d.index + 1][1].toInt()] = dmd[cand[i_d.index + 1]]?.get(i_d.value.toInt())?.get(1)!!
}
val r = toLong(digits, true)
val npr = nmr + 2 * r
if (!isSquare(npr)) {
return
}
count++
print(" R/N %2d:".format(count))
val checkPoint = LocalDateTime.now()
val elapsed = Duration.between(startTime, checkPoint).toMillis()
print(" %9sms".format(elapsed))
val n = toLong(digits, false)
println(" (${commatize(n)})")
rares.add(n)
} else {
for (num in dis[level]) {
di[level] = num
fnpr(cand, di, dis, indicies, nmr, nd, level + 1)
}
}
}
// Recursive closure to generate (n-r) candidates with a given number of digits.
fun fnmr(cand: MutableList<Byte>, list: List<List<Byte>>, indicies: List<List<Byte>>, nd: Int, level: Int) {
if (level == list.size) {
var nmr = 0L
var nmr2 = 0L
for (i_t in allTerms[nd - 2].withIndex()) {
if (cand[i_t.index] >= 0) {
nmr += i_t.value.coeff * cand[i_t.index]
} else {
nmr2 += i_t.value.coeff * -cand[i_t.index]
if (nmr >= nmr2) {
nmr -= nmr2
nmr2 = 0
} else {
nmr2 -= nmr
nmr = 0
}
}
}
if (nmr2 >= nmr) {
return
}
nmr -= nmr2
if (!isSquare(nmr)) {
return
}
val dis = mutableListOf<List<Byte>>()
dis.add(seq(0, ((fml[cand[0]] ?: error("oops")).size - 1).toByte(), 1))
for (i in 1 until cand.size) {
dis.add(seq(0, (dmd[cand[i]]!!.size - 1).toByte(), 1))
}
if (nd % 2 == 1) {
dis.add(il)
}
val di = mutableListOf<Byte>()
for (i in 0 until dis.size) {
di.add(0)
}
fnpr(cand, di, dis, indicies, nmr, nd, 0)
} else {
for (num in list[level]) {
cand[level] = num
fnmr(cand, list, indicies, nd, level + 1)
}
}
}
for (nd in 2..maxDigits) {
digits = mutableListOf()
for (i in 0 until nd) {
digits.add(0)
}
if (nd == 4) {
lists[0].add(zl)
lists[1].add(ol)
lists[2].add(el)
lists[3].add(ol)
} else if (allTerms[nd - 2].size > lists[0].size) {
for (i in 0 until 4) {
lists[i].add(dl)
}
}
val indicies = mutableListOf<List<Byte>>()
for (t in allTerms[nd - 2]) {
indicies.add(listOf(t.ix1, t.ix2))
}
for (list in lists) {
val cand = mutableListOf<Byte>()
for (i in 0 until list.size) {
cand.add(0)
}
fnmr(cand, list, indicies, nd, 0)
}
val checkPoint = LocalDateTime.now()
val elapsed = Duration.between(startTime, checkPoint).toMillis()
println(" %2d digits: %9sms".format(nd, elapsed))
}
rares.sort()
println("\nThe rare numbers with up to $maxDigits digits are:")
for (i_rare in rares.withIndex()) {
println(" %2d: %25s".format(i_rare.index + 1, commatize(i_rare.value)))
}
}</lang>
- Output:
Aggregate timings to process all numbers up to:
R/N 1: 130ms (65)
2 digits: 133ms
3 digits: 133ms
4 digits: 135ms
5 digits: 136ms
R/N 2: 155ms (621,770)
6 digits: 171ms
7 digits: 176ms
8 digits: 238ms
R/N 3: 243ms (281,089,082)
9 digits: 266ms
R/N 4: 272ms (2,022,652,202)
R/N 5: 432ms (2,042,832,002)
10 digits: 693ms
11 digits: 1037ms
R/N 6: 1690ms (872,546,974,178)
R/N 7: 1757ms (872,568,754,178)
R/N 8: 2380ms (868,591,084,757)
12 digits: 2682ms
R/N 9: 3081ms (6,979,302,951,885)
13 digits: 3612ms
R/N 10: 9091ms (20,313,693,904,202)
R/N 11: 9180ms (20,313,839,704,202)
R/N 12: 11322ms (20,331,657,922,202)
R/N 13: 11611ms (20,331,875,722,202)
R/N 14: 12477ms (20,333,875,702,202)
R/N 15: 26933ms (40,313,893,704,200)
R/N 16: 27128ms (40,351,893,720,200)
14 digits: 29696ms
R/N 17: 29759ms (200,142,385,731,002)
R/N 18: 30024ms (221,462,345,754,122)
R/N 19: 33577ms (816,984,566,129,618)
R/N 20: 35392ms (245,518,996,076,442)
R/N 21: 35662ms (204,238,494,066,002)
R/N 22: 35748ms (248,359,494,187,442)
R/N 23: 36108ms (244,062,891,224,042)
R/N 24: 42484ms (403,058,392,434,500)
R/N 25: 42760ms (441,054,594,034,340)
15 digits: 45334ms
R/N 26: 106307ms (2,133,786,945,766,212)
R/N 27: 130390ms (2,135,568,943,984,212)
R/N 28: 134315ms (8,191,154,686,620,818)
R/N 29: 137815ms (8,191,156,864,620,818)
R/N 30: 139449ms (2,135,764,587,964,212)
R/N 31: 141563ms (2,135,786,765,764,212)
R/N 32: 146705ms (8,191,376,864,400,818)
R/N 33: 163353ms (2,078,311,262,161,202)
R/N 34: 204546ms (8,052,956,026,592,517)
R/N 35: 209994ms (8,052,956,206,592,517)
R/N 36: 251686ms (8,650,327,689,541,457)
R/N 37: 254537ms (8,650,349,867,341,457)
R/N 38: 256579ms (6,157,577,986,646,405)
R/N 39: 307145ms (4,135,786,945,764,210)
R/N 40: 347119ms (6,889,765,708,183,410)
16 digits: 350388ms
The rare numbers with up to 16 digits are:
1: 65
2: 621,770
3: 281,089,082
4: 2,022,652,202
5: 2,042,832,002
6: 868,591,084,757
7: 872,546,974,178
8: 872,568,754,178
9: 6,979,302,951,885
10: 20,313,693,904,202
11: 20,313,839,704,202
12: 20,331,657,922,202
13: 20,331,875,722,202
14: 20,333,875,702,202
15: 40,313,893,704,200
16: 40,351,893,720,200
17: 200,142,385,731,002
18: 204,238,494,066,002
19: 221,462,345,754,122
20: 244,062,891,224,042
21: 245,518,996,076,442
22: 248,359,494,187,442
23: 403,058,392,434,500
24: 441,054,594,034,340
25: 816,984,566,129,618
26: 2,078,311,262,161,202
27: 2,133,786,945,766,212
28: 2,135,568,943,984,212
29: 2,135,764,587,964,212
30: 2,135,786,765,764,212
31: 4,135,786,945,764,210
32: 6,157,577,986,646,405
33: 6,889,765,708,183,410
34: 8,052,956,026,592,517
35: 8,052,956,206,592,517
36: 8,191,154,686,620,818
37: 8,191,156,864,620,818
38: 8,191,376,864,400,818
39: 8,650,327,689,541,457
40: 8,650,349,867,341,457
langur
not optimized
It could look something like the following (ignoring whatever optimizations the other examples are using), if it was fast enough. I did not have the time/processor to test finding the first 5. The .israre() function appears to return the right answer, tested with individual numbers.
<lang langur>val .perfectsquare = f isInteger .n ^/ 2
val .israre = f(.n) {
val .r = reverse(.n) if .n == .r: return false val .sum = .n + .r val .diff = .n - .r .diff > 0 and .perfectsquare(.sum) and .perfectsquare(.diff)
}
val .findfirst = f(.max) {
for[=[]] .i = 0; ; .i += 1 {
if .israre(.i) {
_for ~= [.i]
if len(_for) == .max: break
}
}
}
- if you have the time...
writeln "the first 5 rare numbers: ", .findfirst(5)</lang>
With 0.8.11, the built-in reverse() function will flip the digits of a number. Without this, you could write your own function to do so as follows (if not passed any negative numbers).
<lang langur>val .reverse = f toNumber join reverse split .n</lang>
Mathematica/Wolfram Language
<lang Mathematica>c = Compile[[[:Template:K, Integer]],
Module[{out = {0}, start = 0, stop = 0, rlist = {0}, r = 0,
sum = 0.0, diff = 0.0, imax = 8, step = 10},
Do[
If[j == k, imax = 2, imax = 8];
Do[
If[i == 2,
start = i 10^j + 2;
stop = (i + 1) 10^j - 1;
step = 10;
,
start = i 10^j;
stop = (i + 1) 10^j - 1;
step = 1;
];
Do[
rlist = IntegerDigits[n];
r = 0;
Do[
r += rlistri 10^(ri - 1)
,
{ri, 1, Length[rlist]}
];
If[r != n,
sum = n + r;
sum = Sqrt[sum];
If[Floor[sum] == sum,
diff = n - r;
If[diff > 0,
diff = Sqrt[diff];
If[Floor[diff] == diff,
AppendTo[out, n]
]
]
]
]
,
{n, start, stop, step}
]
,
{i, 2, imax, 2}
]
,
{j, 0, k}
];
out
]
];
Rest[c[9]] (*takes about 310 sec*)</lang>
- Output:
{65, 621770, 281089082, 2022652202, 2042832002}
Nim
Translation of the second Go version, limited to 18 digits. <lang Nim>import algorithm, math, strformat, times
type Llst = seq[seq[int]]
const
# Powers of 10.
P = block:
var p: array[19, int64]
p[0] = 1i64
for i in 1..18: p[i] = 10 * p[i - 1]
p
# Digital root lookup array.
Drar = block:
var drar: array[19, int]
for i in 0..18: drar[i] = i shl 1 mod 9
drar
var
d: seq[int] # permutation working slice dac: seq[int] # running digital root slice ac: seq[int64] # accumulator slice pp: seq[int64] # coefficient slice that combines with digits of working slice sr: seq[int64] # temporary list of squares used for building
var
odd = false # flag for odd number of digits sum: int64 # calculated sum of terms (square candidate) cn = 0 # solution counter nd = 2 # number of digits nd1 = nd - 1 # 'nd' helper ln: int # previous value of 'n' (in recurse()) dl: int # length of 'd' slice
func newIntSeq(f, t, s: int): seq[int] =
## Return a sequence of integers. result = newSeq[int]((t - f) div s + 1) var f = f for i in 0..result.high: result[i] = f inc f, s
const
Tlo = @[0, 1, 4, 5, 6] # primary differences starting point All = newIntSeq(-9, 9, 1) # all possible differences Odl = newIntSeq(-9, 9, 2) # odd possible differences Evl = newIntSeq(-8, 8, 2) # even possible differences Thi = @[4, 5, 6, 9, 10, 11, 14, 15, 16] # primary sums starting point Alh = newIntSeq(0, 18, 1) # all possible sums Odh = newIntSeq(1, 17, 2) # odd possible sums Evh = newIntSeq(0, 18, 2) # even possible sums Ten = newIntSeq(0, 9, 1) # used for odd number of digits Z = newIntSeq(0, 0, 1) # no difference, avoids generating a bunch of negative square candidates T7 = @[-3, 7] # shortcut for low 5 Nin = @[9] # shortcut for hi 10 Tn = @[10] # shortcut for hi 0 (unused, unneeded) T12 = @[2, 12] # shortcut for hi 5 O11 = @[1, 11] # shortcut for hi 15 Pos = @[0, 1, 4, 5, 6, 9] # shortcut for 2nd lo 0
var
lul: Llst = @[Z, Odl, @[], @[], Evl, T7, Odl] # shortcut lookup lo primary
luh: Llst = @[Tn, Evh, @[], @[], Evh, T12, Odh, @[], @[],
Evh, Nin, Odh, @[], @[], Odh, O11, Evh] # shortcut lookup hi primary
l2l: Llst = @[Pos, @[], @[], @[], All, @[], All] # shortcut lookup lo secondary
l2h: Llst = @[@[], @[], @[], @[], Alh, @[], Alh, @[], @[],
@[], Alh, @[], @[], @[], Alh, @[], Alh] # shortcut lookup hi secondary
chTen: Llst = @[@[0, 2, 5, 8, 9], @[0, 3, 4, 6, 9], @[1, 4, 7, 8],
@[2, 3, 5, 8], @[0, 3, 6, 7, 9], @[1, 2, 4, 7],
@[2, 5, 6, 8], @[0, 1, 3, 6, 9], @[1, 4, 5, 7]]
chAH: Llst = @[@[0, 2, 5, 8, 9, 11, 14, 17, 18], @[0, 3, 4, 6, 9, 12, 13, 15, 18],
@[1, 4, 7, 8, 10, 13, 16, 17], @[2, 3, 5, 8, 11, 12, 14, 17],
@[0, 3, 6, 7, 9, 12, 15, 16, 18], @[1, 2, 4, 7, 10, 11, 13, 16],
@[2, 5, 6, 8, 11, 14, 15, 17], @[0, 1, 3, 6, 9, 10, 12, 15, 18],
@[1, 4, 5, 7, 10, 13, 14, 16]]
var lu, l2: Llst
func isr(s: int64): int64 {.inline.} =
## Return integer square root. int64(sqrt(float(s)))
proc isRev(nd: int; f, r: int64): bool =
## Recursively determines whether 'r' is the reverse of 'f'. let nd = nd - 1 if f div P[nd] != r mod 10: return false if nd < 1: return true result = isRev(nd, f mod P[nd], r div 10)
proc recurseLE5(lst: Llst; lv: int) =
## Recursive function to evaluate the permutations, no shortcuts.
if lv == dl: # Check if on last stage of permutation.
sum = ac[lv - 1]
if sum > 0:
let rt = int64(sqrt(float(sum)))
if rt * rt == sum: sr.add sum
else:
for n in lst[lv]: # Set up next permutation.
d[lv] = n
if lv == 0: ac[0] = pp[0] * n
else: ac[lv] = ac[lv - 1] + pp[lv] * n # Update accumulated sum.
recurseLE5(lst, lv + 1) # Recursively call next level.
proc recursehi(lst: var Llst; lv: int) =
## Recursive function to evaluate the hi permutations.
## Shortcuts added to avoid generating many non-squares, digital root calc added.
let lv1 = lv - 1
if lv == dl: # Check if on last stage of permutation.
sum = ac[lv1]
if (0x202021202030213 and (1 shl (sum and 63))) != 0:
# Test accumulated sum, append to result if square.
let rt = int64(sqrt(float64(sum)))
if rt * rt == sum: sr.add sum
else:
for n in lst[lv]: # Set up next permutation.
d[lv] = n
if lv == 0:
ac[0] = pp[0] * n
dac[0] = Drar[n] # Update accumulated sum and running dr.
else:
ac[lv] = ac[lv1] + pp[lv] * n
dac[lv] = dac[lv1] + Drar[n]
if dac[lv] > 8: dec dac[lv], 9
case lv # Shortcuts to be performed on designated levels.
of 0: # Primary level: set shortcuts for secondary level.
ln = n
lst[1] = lu[ln]
lst[2] = l2[n]
of 1: # Secondary level: set shortcuts for tertiary level.
case ln # For sums.
of 5, 15: lst[2] = if n < 10: Evh else: Odh
of 9: lst[2] = if (n shr 1 and 1) == 0: Evh else: Odh
of 11: lst[2] = if (n shr 1 and 1) == 1: Evh else: Odh
else: discard
else: discard
if lv == dl - 2:
# Reduce last round according to dr calc.
lst[dl - 1] = if odd: chTen[dac[dl - 2]] else: chAH[dac[dl - 2]]
recursehi(lst, lv + 1) # Recursively call next level.
proc recurselo(lst: var Llst; lv: int) =
## Recursive function to evaluate the lo permutations.
## Shortcuts added to avoid generating many non-squares.
let lv1 = lv - 1
if lv == dl: # Check if on last stage of permutation.
sum = ac[lv1]
if sum > 0:
let rt = int64(sqrt(float64(sum)))
if rt * rt == sum: sr.add sum
else:
for n in lst[lv]: # Set up next permutation.
d[lv] = n
if lv == 0: ac[0] = pp[0] * n
else: ac[lv] = ac[lv1] + pp[lv] * n # Update accumulated sum.
case lv # Shortcuts to be performed on designated levels.
of 0: # Primary level: set shortcuts for secondary level.
ln = n
lst[1] = lu[ln]
lst[2] = l2[n]
of 1: # Secondary level: set shortcuts for tertiary level.
case ln # For difs.
of 1: lst[2] = if ((n + 9) shr 1 and 1) == 0: Evl else: Odl
of 5: lst[2] = if n < 0: Evl else: Odl
else: discard
else: discard
recurselo(lst, lv + 1) # Recursively call next level.
proc listEm(lst: var Llst; plu, pl2: Llst): seq[int64] =
## Produces a list of candidate square numbers. dl = lst.len d = newSeq[int](dl) sr.setLen(0) lu = plu l2 = pl2 ac = newSeq[int64](dl) dac = newSeq[int](dl) pp = newSeq[int64](dl) # Build coefficients array. for i in 0..<dl: pp[i] = if lst[0].len > 6: P[nd1 - i] + P[i] else: P[nd1 - i] - P[i] # Call appropriate recursive function. if nd <= 5: recurseLE5(lst, 0) elif lst[0].len > 8: recursehi(lst, 0) else: recurselo(lst, 0) result = sr
proc reveal(lo, hi: openArray[int64]) =
## Reveal whether combining two lists of squares can produce a rare number.
var s: seq[string] # Temporary list of results.
for l in lo:
for h in hi:
let r = (h - l) shr 1
let f = h - r # Generate all possible fwd & rev candidates from lists.
if isRev(nd, f, r):
s.add &"{f:20} {isr(h):11} {isr(l):10} "
s.sort()
if s.len > 0:
for t in s:
inc cn
let tt = if t != s[^1]: "\n" else: ""
stdout.write &"{cn:2} {t}{tt}"
else:
stdout.write &"{\"\":48}"
func formatTime(d: Duration): string =
var f = d.inMilliseconds
var s = f div 1000
f = f mod 1000
var m = s div 60
s = s mod 60
let h = m div 60
m = m mod 60
result = &"{h:02}:{m:02}:{s:02}.{f:03}"
var
lls: Llst = @[Tlo] hls: Llst = @[Thi]
var bstart, tstart = now()
echo &"""nth {"forward":>19} {"rt.sum":>11} {"rt.dif":>10} digs {"block time":>11} {"total time":>13}"""
while nd <= 18:
if nd > 2:
if odd:
hls.add Ten
else:
lls.add All
hls[^1] = Alh
reveal(listEm(lls, lul, l2l), listEm(hls, luh, l2h))
if not odd and nd > 5:
# Restore last element of hls, so that dr shortcut doesn't mess up next nd.
hls[^1] = Alh
let bTime = formatTime(now() - bstart)
let tTime = formatTime(now() - tstart)
echo &"{nd:2}: {bTime} {tTime}"
bstart = now() # Restart block timing.
nd1 = nd
inc nd
odd = not odd</lang>
- Output:
nth forward rt.sum rt.dif digs block time total time
1 65 11 3 2: 00:00:00.000 00:00:00.000
3: 00:00:00.000 00:00:00.000
4: 00:00:00.000 00:00:00.000
5: 00:00:00.000 00:00:00.000
2 621770 836 738 6: 00:00:00.000 00:00:00.000
7: 00:00:00.000 00:00:00.001
8: 00:00:00.001 00:00:00.002
3 281089082 23708 330 9: 00:00:00.002 00:00:00.005
4 2022652202 63602 300
5 2042832002 63602 6360 10: 00:00:00.005 00:00:00.010
11: 00:00:00.040 00:00:00.051
6 868591084757 1275175 333333
7 872546974178 1320616 32670
8 872568754178 1320616 33330 12: 00:00:00.066 00:00:00.117
9 6979302951885 3586209 1047717 13: 00:00:00.486 00:00:00.603
10 20313693904202 6368252 269730
11 20313839704202 6368252 270270
12 20331657922202 6368252 329670
13 20331875722202 6368252 330330
14 20333875702202 6368252 336330
15 40313893704200 6368252 6330336
16 40351893720200 6368252 6336336 14: 00:00:00.968 00:00:01.572
17 200142385731002 20006998 69300
18 204238494066002 20122102 1891560
19 221462345754122 21045662 69300
20 244062891224042 22011022 1908060
21 245518996076442 22140228 921030
22 248359494187442 22206778 1891560
23 403058392434500 20211202 19940514
24 441054594034340 22011022 19940514
25 816984566129618 40421606 250800 15: 00:00:09.416 00:00:10.989
26 2078311262161202 64030648 7529850
27 2133786945766212 65272218 2666730
28 2135568943984212 65272218 3267330
29 2135764587964212 65272218 3326670
30 2135786765764212 65272218 3333330
31 4135786945764210 65272218 63333336
32 6157577986646405 105849161 33333333
33 6889765708183410 83866464 82133718
34 8052956026592517 123312255 29999997
35 8052956206592517 123312255 30000003
36 8191154686620818 127950856 3299670
37 8191156864620818 127950856 3300330
38 8191376864400818 127950856 3366330
39 8650327689541457 127246955 33299667
40 8650349867341457 127246955 33300333 16: 00:00:18.483 00:00:29.472
41 22542040692914522 212329862 333300
42 67725910561765640 269040196 251135808
43 86965750494756968 417050956 33000 17: 00:02:49.293 00:03:18.766
44 225342456863243522 671330638 297000
45 225342458663243522 671330638 303000
46 225342478643243522 671330638 363000
47 284684666566486482 754565658 30000
48 284684868364486482 754565658 636000
49 297128548234950692 770186978 32697330
50 297128722852950692 770186978 32702670
51 297148324656930692 770186978 33296670
52 297148546434930692 770186978 33303330
53 497168548234910690 770186978 633363336
54 619431353040136925 1071943279 299667003
55 619631153042134925 1071943279 300333003
56 631688638047992345 1083968809 297302703
57 633288858025996145 1083968809 302637303
58 633488632647994145 1083968809 303296697
59 653488856225994125 1083968809 363303363
60 811865096390477018 1273828556 33030330
61 865721270017296468 1315452006 32071170
62 871975098681469178 1320582934 3303300
63 898907259301737498 1339270086 64576740 18: 00:05:45.616 00:09:04.383
Perl
<lang perl>#!/usr/bin/perl
use strict; # https://rosettacode.org/wiki/Rare_numbers use warnings; use integer;
my $count = 0; my @squares; for my $large ( 0 .. 1e5 )
{
my $largesquared = $squares[$large] = $large * $large; # $large ** 2;
for my $small ( 0 .. $large - 1 )
{
my $n = $largesquared + $squares[$small];
2 * $large * $small == reverse $n or next;
printf "%12s %s\n", $n, scalar reverse $n;
$n == reverse $n and die "oops!"; # palindrome check
++$count >= 5 and exit;
}
}</lang>
- Output:
65 56
621770 077126
281089082 280980182
2022652202 2022562202
2042832002 2002382402
Phix
naive
Ridiculously slow, 90s just for the first 3. <lang Phix>function revn(atom n, integer nd)
atom r = 0
for i=1 to nd do
r = r*10+remainder(n,10)
n = floor(n/10)
end for
return r
end function
integer nd = 2, count = 0 atom lim = 99, n = 9, t0 = time() while true do
n += 1
atom r = revn(n,nd)
if r<n then
atom s = n+r,
d = n-r
if s=power(floor(sqrt(s)),2)
and d=power(floor(sqrt(d)),2) then
count += 1
?{count,n,elapsed(time()-t0)}
if count=3 then exit end if
end if
end if
if n=lim then
-- ?{"lim",lim,elapsed(time()-t0)}
lim = lim*10+9
nd += 1
end if
end while</lang>
- Output:
{1,65,"0s"}
{2,621770,"0.2s"}
{3,281089082,"1 minute and 29s"}
advanced
Quite slow: over 10 mins for first 25, vs 53 secs of Go... easily the worst such (ie Phix vs. Go) comparison I have yet seen.
This task exposes some of the weaknesses of Phix, specifically subscripting speed (suprisingly not usually that much of an issue),
and the fact that functions are never inlined. The relatively innoculous-looking and dirt simple to_atom() routine is responsible
for over 30% of the run time. Improvements welcome: run p -d test, examine the list.asm produced from this source, and discuss or
make suggestions on my talk page.
<lang Phix>constant maxDigits = 15
enum COEFF, TDXA, TDXB -- struct term = {atom coeff, integer idxa, idxb}
-- (see allTerms below)
integer nd, -- number of digits
count -- of solutions found earlier, for lower nd
sequence rares -- (cleared after sorting/printing for each nd)
function to_atom(sequence digits) -- convert digits array to an atom value
atom r = 0
for i=1 to length(digits) do
r = r * 10 + digits[i]
end for
return r
end function
-- psq eliminates 52 out of 64 of numbers fairly cheaply, which translates -- to approximately 66% of numbers, or around 10% off the overall time. -- NB: only tested to 9,007,199,254,740,991, then again I found no more new -- bit patterns after just 15^2.
constant psq = int_to_bits(#02030213,32)& -- #0202021202030213 --> bits,
int_to_bits(#02020212,32) -- in 32/64-bit compatible way.
function isSquare(atom n) -- determine if n is a perfect square or not
if psq[and_bits(n,63)+1] then
atom r = floor(sqrt(n))
return r * r = n
end if
return false
end function
procedure fnpr(integer level, atom nmr, sequence di, dis, candidates, indices, fml, dmd) -- generate (n+r) candidates from (n-r) candidates
if level>length(dis) then
sequence digits = repeat(0,nd)
-- (the precise why of how this populates digits has eluded me...)
integer {a,b} = indices[1],
c = candidates[1]+1,
d = di[1]+1
digits[a] = fml[c][d][1]
digits[b] = fml[c][d][2]
integer le = length(di)
if remainder(nd,2) then
d = floor(nd/2)+1
digits[d] = di[le]
le -= 1
end if
for dx=2 to le do
{a,b} = indices[dx]
c = candidates[dx]+10
d = di[dx]+1
digits[a] = dmd[c][d][1]
digits[b] = dmd[c][d][2]
end for
atom npr = nmr + to_atom(reverse(digits))*2 -- (npr == 'n + r')
if isSquare(npr) then
rares &= to_atom(digits)
-- (note this gets overwritten by sorted set:)
printf(1,"working... %2d: %,d\r", {count+length(rares),rares[$]})
end if
else
for n=0 to dis[level] do
di[level] = n
fnpr(level+1, nmr, di, dis, candidates, indices, fml, dmd)
end for
end if
end procedure
procedure fnmr(sequence terms, list, candidates, indices, fml, dmd, integer level) -- generate (n-r) candidates with a given number of digits.
if level>length(list) then
atom nmr = 0 -- (nmr == 'n - r')
for i=1 to length(terms) do
nmr += terms[i][COEFF] * candidates[i]
end for
if nmr>0 and isSquare(nmr) then
integer c = candidates[1]+1,
l = length(fml[c])-1
sequence dis = {l}
for i=2 to length(candidates) do
c = candidates[i]+10
l = length(dmd[c])-1
dis = append(dis,l)
end for
if remainder(nd,2) then dis = append(dis,9) end if
-- (above generates dis of eg {1,4,7,9} for nd=7, which as far
-- as I (lightly) understand it scans for far fewer candidate
-- pairs than a {9,9,9,9} would, or something like that.)
sequence di = repeat(0,length(dis))
-- (di is the current "dis-scan", eg {0,0,0,0} to {1,4,7,9})
fnpr(1, nmr, di, dis, candidates, indices, fml, dmd)
end if
else
for n=1 to length(list[level]) do
candidates[level] = list[level][n]
fnmr(terms, list, candidates, indices, fml, dmd, level+1)
end for
end if
end procedure
constant dl = tagset(9,-9), -- all differences (-9..+9 by 1)
zl = tagset(0, 0), -- zero difference (0 only)
el = tagset(8,-8, 2), -- even differences (-8 to +8 by 2)
ol = tagset(9,-9, 2), -- odd differences (-9..+9 by 2)
il = tagset(9, 0) -- all integers (0..9 by 1)
procedure main()
atom start = time()
-- terms of (n-r) expression for number of digits from 2 to maxdigits
sequence allTerms = {}
atom pow = 1
for r=2 to maxDigits do
sequence terms = {}
pow *= 10
atom p1 = pow, p2 = 1
integer tdxa = 0, tdxb = r-1
while tdxa < tdxb do
terms = append(terms,{p1-p2, tdxa, tdxb}) -- {COEFF,TDXA,TDXB}
p1 /= 10
p2 *= 10
tdxa += 1
tdxb -= 1
end while
allTerms = append(allTerms,terms)
end for
--/* --(This is what the above loop creates:) --pp(allTerms,{pp_Nest,1,pp_StrFmt,3,pp_IntCh,false,pp_IntFmt,"%d",pp_FltFmt,"%d",pp_Maxlen,148}) {Template:9,0,1,
Template:99,0,2, {{999,0,3}, {90,1,2}}, {{9999,0,4}, {990,1,3}}, {{99999,0,5}, {9990,1,4}, {900,2,3}}, {{999999,0,6}, {99990,1,5}, {9900,2,4}}, {{9999999,0,7}, {999990,1,6}, {99900,2,5}, {9000,3,4}}, {{99999999,0,8}, {9999990,1,7}, {999900,2,6}, {99000,3,5}}, {{999999999,0,9}, {99999990,1,8}, {9999900,2,7}, {999000,3,6}, {90000,4,5}}, {{9999999999,0,10}, {999999990,1,9}, {99999900,2,8}, {9999000,3,7}, {990000,4,6}}, {{99999999999,0,11}, {9999999990,1,10}, {999999900,2,9}, {99999000,3,8}, {9990000,4,7}, {900000,5,6}}, {{999999999999,0,12}, {99999999990,1,11}, {9999999900,2,10}, {999999000,3,9}, {99990000,4,8}, {9900000,5,7}}, {{9999999999999,0,13}, {999999999990,1,12}, {99999999900,2,11}, {9999999000,3,10}, {999990000,4,9}, {99900000,5,8}, {9000000,6,7}}, {{99999999999999,0,14}, {9999999999990,1,13}, {999999999900,2,12}, {99999999000,3,11}, {9999990000,4,10}, {999900000,5,9}, {99000000,6,8}}}
--*/
-- map of first minus last digits for 'n' to pairs giving this value
sequence fml = repeat({},10) -- (aka 0..9)
-- (fml == 'first minus last')
fml[1] = {{2, 2}, {8, 8}}
fml[2] = {{6, 5}, {8, 7}}
fml[5] = Template:4, 0
-- fml[6] = Template:8, 3 -- (um? - needs longer lists, & that append(lists[4],dl) below)
fml[7] = {{6, 0}, {8, 2}}
-- sequence lists = {Template:0,Template:1,Template:4,Template:5,Template:6}
sequence lists = {Template:0,Template:1,Template:4,Template:6}
-- map of other digit differences for 'n' to pairs giving this value
sequence dmd = repeat({},19) -- (aka -9..+9, so add 10 when indexing dmd)
-- (dmd == 'digit minus digit')
for tens=0 to 9 do
integer d = tens+10
for ones=0 to 9 do
dmd[d] = append(dmd[d], {tens,ones})
d -= 1
end for
end for
--/* --(This is what the above loop creates:) --pp(dmd,{pp_Nest,1,pp_StrFmt,3,pp_IntCh,false}) {Template:0,9,
{{0,8}, {1,9}},
{{0,7}, {1,8}, {2,9}},
{{0,6}, {1,7}, {2,8}, {3,9}},
{{0,5}, {1,6}, {2,7}, {3,8}, {4,9}},
{{0,4}, {1,5}, {2,6}, {3,7}, {4,8}, {5,9}},
{{0,3}, {1,4}, {2,5}, {3,6}, {4,7}, {5,8}, {6,9}},
{{0,2}, {1,3}, {2,4}, {3,5}, {4,6}, {5,7}, {6,8}, {7,9}},
{{0,1}, {1,2}, {2,3}, {3,4}, {4,5}, {5,6}, {6,7}, {7,8}, {8,9}},
{{0,0}, {1,1}, {2,2}, {3,3}, {4,4}, {5,5}, {6,6}, {7,7}, {8,8}, {9,9}},
{{1,0}, {2,1}, {3,2}, {4,3}, {5,4}, {6,5}, {7,6}, {8,7}, {9,8}},
{{2,0}, {3,1}, {4,2}, {5,3}, {6,4}, {7,5}, {8,6}, {9,7}},
{{3,0}, {4,1}, {5,2}, {6,3}, {7,4}, {8,5}, {9,6}},
{{4,0}, {5,1}, {6,2}, {7,3}, {8,4}, {9,5}},
{{5,0}, {6,1}, {7,2}, {8,3}, {9,4}},
{{6,0}, {7,1}, {8,2}, {9,3}},
{{7,0}, {8,1}, {9,2}},
{{8,0}, {9,1}},
Template:9,0}
--*/
count = 0
printf(1,"digits time nth rare numbers:\n")
nd = 2
while nd <= maxDigits do
rares = {}
sequence terms = allTerms[nd-1]
if nd=4 then
lists[1] = append(lists[1],zl)
lists[2] = append(lists[2],ol)
lists[3] = append(lists[3],el)
-- lists[4] = append(lists[4],dl) -- if fml[6] = Template:8, 3 -- lists[5] = append(lists[5],ol) -- ""
lists[4] = append(lists[4],ol) -- else
elsif length(terms)>length(lists[1]) then
for i=1 to length(lists) do
lists[i] = append(lists[i],dl)
end for
end if
sequence indices = {}
for t=1 to length(terms) do
sequence term = terms[t]
-- (we may as well make this 1-based while here)
indices = append(indices,{term[TDXA]+1,term[TDXB]+1})
end for
for i=1 to length(lists) do
sequence list = lists[i],
candidates = repeat(0,length(list))
fnmr(terms, list, candidates, indices, fml, dmd, 1)
end for
-- (re-)output partial results for this nd-set in sorted order:
rares = sort(rares)
for i=1 to length(rares) do
count += 1
printf(1,"%12s %2d: %,19d \n", {"",count,rares[i]})
end for
printf(1," %2d %5s\n", {nd, elapsed_short(time()-start)})
nd += 1
end while
end procedure main()</lang>
- Output:
digits time nth rare numbers:
1: 65
2 0s
3 0s
4 0s
5 0s
2: 621,770
6 0s
7 0s
8 0s
3: 281,089,082
9 0s
4: 2,022,652,202
5: 2,042,832,002
10 0s
11 1s
6: 868,591,084,757
7: 872,546,974,178
8: 872,568,754,178
12 15s
9: 6,979,302,951,885
13 29s
10: 20,313,693,904,202
11: 20,313,839,704,202
12: 20,331,657,922,202
13: 20,331,875,722,202
14: 20,333,875,702,202
15: 40,313,893,704,200
16: 40,351,893,720,200
14 6:07
17: 200,142,385,731,002
18: 204,238,494,066,002
19: 221,462,345,754,122
20: 244,062,891,224,042
21: 245,518,996,076,442
22: 248,359,494,187,442
23: 403,058,392,434,500
24: 441,054,594,034,340
25: 816,984,566,129,618
15 10:42
Python
naive
A simple implementation, just to show how elegant python can be. Searches up to 10^11 in about 3hours w/ pypy on my Intel Core i5.
<lang Python>
- rare.py
- find rare numbers
- by kHz
from math import floor, sqrt from datetime import datetime
def main(): start = datetime.now() for i in xrange(1, 10 ** 11): if rare(i): print "found a rare:", i end = datetime.now() print "time elapsed:", end - start
def is_square(n): s = floor(sqrt(n + 0.5)) return s * s == n
def reverse(n): return int(str(n)[::-1])
def is_palindrome(n): return n == reverse(n)
def rare(n): r = reverse(n) return ( not is_palindrome(n) and n > r and is_square(n+r) and is_square(n-r) )
if __name__ == '__main__': main()
</lang>
Quackery
Naive version.
Timings for Quackery running on PyPy3 on one core of a 3 GHz 6-Core Intel Core i5 Mac mini at 100%.
First result; less than a second. Second result; less than five minutes (possibly a lot less - I went to make a cup of tea), third result; more than twelve and less than fifteen hours. At this point I estimated from other timings on this page that getting the fourth and fifth results would take about a week and killed the process. :-(
On the other hand; code development time, less than five minutes. :-)
<lang Quackery>[ dup 1
[ 2dup > while
+ 1 >>
2dup / again ]
drop nip ] is sqrt ( n --> n )
[ dup sqrt 2 ** = not ] is !square ( n --> b )
[ number$ reverse
$->n drop ] is revnumber ( n --> n )
[ 0 swap
[ base share /mod
rot + swap
dup 0 = until ]
drop ] is digitalroot ( n --> n )
[ true swap
dup revnumber
2dup > not iff
[ 2drop not ] done
2dup + !square iff
[ 2drop not ] done
2dup - !square iff
[ 2drop not ] done
2drop ] is rare ( n --> b )
[ 0
[ 1+ dup rare if
[ dup echo cr
dip [ 1 - ] ]
over 0 = until ]
2drop ] is echorarenums ( n --> b )
5 echorarenums</lang>
- Output:
5 621770 281089082
Raku
Translation of Rust
<lang perl6># 20220315 Raku programming solution
sub rare (\target where ( target > 0 and target ~~ Int )) {
my \digit = $ = 2; my $count = 0; my @numeric_digits = 0..9 Z, 0 xx *; my @diffs1 = 0,1,4,5,6;
# all possible digits pairs to calculate potential diffs my @pairs = 0..9 X 0..9; my @all_diffs = -9..9;
# lookup table for the first diff
my @lookup_1 = [ [[2, 2], [8, 8]], # Diff = 0
[[8, 7], [6, 5]], # Diff = 1
[],
[],
[[4, 0], ], # Diff = 4
[[8, 3], ], # Diff = 5
[[6, 0], [8, 2]], ]; # Diff = 6
# lookup table for all the remaining diffs
given my %lookup_n { for @pairs -> \pair { $_{ [-] pair.values }.push: pair } }
loop {
my @powers = 10 <<**<< (0..digit-1); # powers like 1, 10, 100, 1000....
# for n-r (aka L) the required terms, like 9/ 99 / 999 & 90 / 99999 & 9999 & 900 etc
my @terms = (@powers.reverse Z- @powers).grep: * > 0 ;
# create a cartesian product for all potential diff numbers
# for the first use the very short one, for all other the complete 19 element
my @diff_list = digit == 2 ?? @diffs1 !! [X] @diffs1, |(@all_diffs xx digit div 2 - 1);
my @diff_list_iter = gather for @diff_list -> \k {
# remove invalid first diff/second diff combinations
{ take k andthen next } if k.elems == 1 ;
given (my (\a,\b) = k.values) {
when a == 0 && b != 0 { next }
when a == 1 && b ∉ [ -7, -5, -3, -1, 1, 3, 5, 7 ] { next }
when a == 4 && b ∉ [ -8, -6, -4, -2, 0, 2, 4, 6, 8 ] { next }
when a == 5 && b ∉ [ -3, 7 ] { next }
when a == 6 && b ∉ [ -9, -7, -5, -3, -1, 1, 3, 5, 7, 9 ] { next }
default { take k }
}
}
for @diff_list_iter -> \diffs {
# calculate difference of original n and its reverse (aka L = n-r)
# which must be a perfect square
if (my \L = [+] diffs <<*>> @terms) > 0 and { $_ == $_.Int }(L.sqrt) {
# potential candiate, at least L is a perfect square
# placeholder for the digits
my \dig = @ = 0 xx digit;
# generate a cartesian product for each identified diff using the lookup tables
my @c_iter = digit == 2
?? @lookup_1[diffs[0]].map: { [ $_ ] }
!! [X] @lookup_1[diffs[0]], |(1..(+diffs + (digit % 2 - 1))).map: -> \k {
k == diffs ?? @numeric_digits !! %lookup_n{diffs[k]} }
# check each H (n+r) by using digit combination
for @c_iter -> \elt {
for elt.kv -> \i, \pair { dig[i,digit-1-i] = pair.values }
# for numbers with odd # digits restore the middle digit
# which has been overwritten at the end of the previous cycle
dig[(digit - 1) div 2] = elt[+elt - 1][0] if digit % 2 == 1 ;
my \rev = ( my \num = [~] dig ).flip;
if num > rev and { $_ == $_.Int }((num+rev).sqrt) {
printf "%d: %12d reverse %d\n", $count+1, num, rev;
exit if ++$count == target;
}
}
}
}
digit++
}
}
my $N = 5; say "The first $N rare numbers are,"; rare $N;</lang>
- Output:
The first 5 rare numbers are, 1: 65 reverse 56 2: 621770 reverse 77126 3: 281089082 reverse 280980182 4: 2022652202 reverse 2022562202 5: 2042832002 reverse 2002382402
Using NativeCall with Rust
This example will make use of a modified version of the 'advanced' routine from the Rust entry. <lang bash>~> cargo new --lib Rare && cd $_
Created library `Rare` package</lang>
Add to the stock manifest file, Cargo.toml, all required dependencies and build target, <lang bash>~/Rare> tail -5 Cargo.toml [dependencies] itertools = "0.10.3"
[lib] crate-type = ["cdylib"]</lang> Now replace the src/lib.rs file with the following,
lib.rs <lang rust>use itertools::Itertools; use std::collections::HashMap;
fn isqrt(n: u64) -> u64 {
let mut s = (n as f64).sqrt() as u64;
s = (s + n / s) >> 1;
if s * s > n {
s - 1
} else {
s
}
}
fn is_square(n: u64) -> bool {
match n & 0xf {
0 | 1 | 4 | 9 => {
let t = isqrt(n);
t * t == n
}
_ => false,
}
}
- [no_mangle]
/// This algorithm uses an advanced search strategy based on Nigel Galloway's approach pub extern "C" fn advanced64(target: u8) -> *mut u64 {
// setup let digit = 2u8; let mut results = Vec::new(); let mut counter = 0_u8;
let numeric_digits = (0..=9).map(|x| [x, 0]).collect::<Vec<_>>(); let diffs1: Vec<i8> = vec![0, 1, 4, 5, 6];
// all possible digits pairs to calculate potential diffs
let pairs = (0_i8..=9)
.cartesian_product(0_i8..=9)
.map(|x| [x.0, x.1])
.collect::<Vec<_>>();
let all_diffs = (-9i8..=9).collect::<Vec<_>>();
// lookup table for the first diff
let lookup_1 = vec![
vec![[2, 2], [8, 8]], //Diff = 0
vec![[8, 7], [6, 5]], //Diff = 1
vec![],
vec![],
vec!4, 0, // Diff = 4
vec!8, 3, // Diff = 5
vec![[6, 0], [8, 2]], // Diff = 6
];
// lookup table for all the remaining diffs let lookup_n: HashMap<i8, Vec<_>> = pairs.into_iter().into_group_map_by(|elt| elt[0] - elt[1]);
let mut d = digit;
while target > counter {
// powers like 1, 10, 100, 1000....
let powers = (0..d).map(|x| 10_u64.pow(x.into())).collect::<Vec<u64>>();
// for n-r (aka L) the required terms, like 9/ 99 / 999 & 90 / 99999 & 9999 & 900 etc
let terms = powers
.iter()
.zip(powers.iter().rev())
.map(|(a, b)| b.checked_sub(*a).unwrap_or(0))
.filter(|x| *x != 0)
.collect::<Vec<u64>>();
// create a cartesian product for all potential diff numbers
// for the first use the very short one, for all other the complete 19 element
let diff_list_iter = (0_u8..(d / 2))
.map(|i| match i {
0 => diffs1.iter(),
_ => all_diffs.iter(),
})
.multi_cartesian_product()
// remove invalid first diff/second diff combinations - custom iterator would be probably better
.filter(|x| {
if x.len() == 1 {
return true;
}
match (*x[0], *x[1]) {
(a, b) if (a == 0 && b != 0) => false,
(a, b) if (a == 1 && ![-7, -5, -3, -1, 1, 3, 5, 7].contains(&b)) => false,
(a, b) if (a == 4 && ![-8, -6, -4, -2, 0, 2, 4, 6, 8].contains(&b)) => false,
(a, b) if (a == 5 && ![7, -3].contains(&b)) => false,
(a, b) if (a == 6 && ![-9, -7, -5, -3, -1, 1, 3, 5, 7, 9].contains(&b)) => {
false
}
_ => true,
}
});
'OUTER: for diffs in diff_list_iter {
// calculate difference of original n and its reverse (aka L = n-r)
// which must be a perfect square
let l: i64 = diffs
.iter()
.zip(terms.iter())
.map(|(diff, term)| **diff as i64 * *term as i64)
.sum();
if l > 0 && is_square(l.try_into().unwrap()) {
// potential candiate, at least L is a perfect square
// placeholder for the digits
let mut dig: Vec<i8> = vec![0_i8; d.into()];
// generate a cartesian product for each identified diff using the lookup tables
let c_iter = (0..(diffs.len() + d as usize % 2))
.map(|i| match i {
0 => lookup_1[*diffs[0] as usize].iter(),
_ if i != diffs.len() => lookup_n.get(diffs[i]).unwrap().iter(),
_ => numeric_digits.iter(), // for the middle digits
})
.multi_cartesian_product();
// check each H (n+r) by using digit combination
c_iter.for_each(|elt| {
for (i, digit_pair) in elt.iter().enumerate() {
dig[i] = digit_pair[0];
dig[d as usize - 1 - i] = digit_pair[1]
}
// for numbers with odd # digits restore the middle digit
// which has been overwritten at the end of the previous cycle
if d % 2 == 1 {
dig[(d as usize - 1) / 2] = elt[elt.len() - 1][0];
}
let num = dig
.iter()
.rev()
.enumerate()
.fold(0_u64, |acc, (i, d)| acc + 10_u64.pow(i as u32) * *d as u64);
let reverse = dig
.iter()
.enumerate()
.fold(0_u64, |acc, (i, d)| acc + 10_u64.pow(i as u32) * *d as u64);
if num > reverse && is_square(num + reverse) {
counter += 1;
results.push(num);
}
});
if counter == target {
break 'OUTER;
}
}
}
d += 1
}
let ptr = results.as_mut_ptr();
std::mem::forget(results); // circumvent the destructor
ptr
}</lang> The needful shared library will be available after the build command, <lang bash>~/Rare> cargo build ~/Rare> file target/debug/libRare.so target/debug/libRare.so: ELF 64-bit LSB shared object, x86-64, version 1 (SYSV), dynamically linked, BuildID[sha1]=4f904cce7f8e82130826bf46f93fe9fe944ab9d0, with debug_info, not stripped</lang> Here is the main Raku program, <lang perl6> use NativeCall;
constant LIB = '/home/hkdtam/Rare/target/debug/libRare.so';
sub advanced64(uint8) returns Pointer[uint64] is native(LIB) {*}
my $N = 5; say "The first $N rare numbers are,";
for (advanced64 $N)[^$N].kv -> \nth,\rare {
printf "%d: %12d reverse %d\n", nth+1, { $_, $_.flip }(rare)
}</lang> Output is the same.
REXX
(See the discussion page for a simplistic 1st version that computes rare numbers only using the task's basic rules).
Most of the hints (properties of rare numbers) within Shyam Sunder Gupta's webpage have been incorporated in this
REXX program and the logic is now expressed within the list of AB...PQ (abutted numbers within the @g list).
These improvements made this REXX version around 25% faster than the previous version (see the discussion page). <lang rexx>/*REXX program calculates and displays a specified amount of rare numbers. */ numeric digits 20; w= digits() + digits() % 3 /*use enough dec. digs for calculations*/ parse arg many . /*obtain optional argument from the CL.*/ if many== | many=="," then many= 5 /*Not specified? Then use the default.*/ @g= 2002 2112 2222 2332 2442 2552 2662 2772 2882 2992 4000 4010 4030 4050 4070 4090 4100 ,
4110 4120 4140 4160 4180 4210 4230 4250 4270 4290 4300 4320 4340 4360 4380 4410 4430 , 4440 4450 4470 4490 4500 4520 4540 4560 4580 4610 4630 4650 4670 4690 4700 4720 4740 , 4760 4780 4810 4830 4850 4870 4890 4900 4920 4940 4960 4980 4990 6010 6015 6030 6035 , 6050 6055 6070 6075 6090 6095 6100 6105 6120 6125 6140 6145 6160 6165 6180 6185 6210 , 6215 6230 6235 6250 6255 6270 6275 6290 6295 6300 6305 6320 6325 6340 6345 6360 6365 , 6380 6385 6410 6415 6430 6435 6450 6455 6470 6475 6490 6495 6500 6505 6520 6525 6540 , 6545 6560 6565 6580 6585 6610 6615 6630 6635 6650 6655 6670 6675 6690 6695 6700 6705 , 6720 6725 6740 6745 6760 6765 6780 6785 6810 6815 6830 6835 6850 6855 6870 6875 6890 , 6895 6900 6905 6920 6925 6940 6945 6960 6965 6980 6985 8007 8008 8017 8027 8037 8047 , 8057 8067 8077 8087 8092 8097 8107 8117 8118 8127 8137 8147 8157 8167 8177 8182 8187 , 8197 8228 8272 8297 8338 8362 8387 8448 8452 8477 8542 8558 8567 8632 8657 8668 8722 , 8747 8778 8812 8837 8888 8902 8927 8998 /*4 digit abutted numbers for AB and PQ*/
@g#= words(@g)
/* [↓]─────────────────boolean arrays are used for checking for digit presence.*/
@dr.=0; @dr.2= 1; @dr.5=1 ; @dr.8= 1; @dr.9= 1 /*rare # must have these digital roots.*/ @ps.=0; @ps.2= 1; @ps.3= 1; @ps.7= 1; @ps.8= 1 /*perfect squares must end in these.*/ @149.=0; @149.1=1; @149.4=1; @149.9=1 /*values for Z that need an even Y. */ @odd.=0; do i=-9 by 2 to 9; @odd.i=1 /* " " N " " " " A. */
end /*i*/
@gen.=0; do i=1 for words(@g); parse value word(@g,i) with a 2 b 3 p 4 q; @gen.a.b.p.q=1
/*# AB···PQ could be a good rare value*/
end /*i*/
div9= 9 /*dif must be ÷ 9 when N has even #digs*/ evenN= \ (10 // 2) /*initial value for evenness of N. */
- = 0 /*the number of rare numbers (so far)*/
do n=10 /*Why 10? All 1 dig #s are palindromic*/
parse var n a 2 b 3 -2 p +1 q /*get 1st\2nd\penultimate\last digits. */
if @odd.a then do; n=n+10**(length(n)-1)-1 /*bump N so next N starts with even dig*/
evenN=\(length(n+1)//2) /*flag when N has an even # of digits. */
if evenN then div9= 9 /*when dif isn't divisible by 9 ... */
else div9= 99 /* " " " " " 99 " */
iterate /*let REXX do its thing with DO loop.*/
end /* {it's allowed to modify a DO index} */
if \@gen.a.b.p.q then iterate /*can N not be a rare AB···PQ number?*/
r= reverse(n) /*obtain the reverse of the number N. */
if r>n then iterate /*Difference will be negative? Skip it*/
if n==r then iterate /*Palindromic? Then it can't be rare.*/
dif= n-r; parse var dif -2 y +1 z /*obtain the last 2 digs of difference.*/
if @ps.z then iterate /*Not 0, 1, 4, 5, 6, 9? Not perfect sq.*/
select
when z==0 then if y\==0 then iterate /*Does Z = 0? Then Y must be zero. */
when z==5 then if y\==2 then iterate /*Does Z = 5? Then Y must be two. */
when z==6 then if y//2==0 then iterate /*Does Z = 6? Then Y must be odd. */
otherwise if @149.z then if y//2 then iterate /*Z=1,4,9? Y must be even*/
end /*select*/ /* [↑] the OTHERWISE handles Z=8 case.*/
if dif//div9\==0 then iterate /*Difference isn't ÷ by div9? Then skip*/
sum= n+r; parse var sum -2 y +1 z /*obtain the last two digits of the sum*/
if @ps.z then iterate /*Not 0, 2, 5, 8, or 9? Not perfect sq.*/
select
when z==0 then if y\==0 then iterate /*Does Z = 0? Then Y must be zero. */
when z==5 then if y\==2 then iterate /*Does Z = 5? Then Y must be two. */
when z==6 then if y//2==0 then iterate /*Does Z = 6? Then Y must be odd. */
otherwise if @149.z then if y//2 then iterate /*Z=1,4,9? Y must be even*/
end /*select*/ /* [↑] the OTHERWISE handles Z=8 case.*/
if evenN then if sum//11 \==0 then iterate /*N has even #digs? Sum must be ÷ by 11*/
$= a + b /*a head start on figuring digital root*/
do k=3 for length(n) - 2 /*now, process the rest of the digits. */
$= $ + substr(n, k, 1) /*add the remainder of the digits in N.*/
end /*k*/
do while $>9 /* [◄] Algorithm is good for 111 digs.*/
if $>9 then $= left($,1) + substr($,2,1) + substr($,3,1,0) /*>9? Reduce it.*/
end /*while*/
if \@dr.$ then iterate /*Doesn't have good digital root? Skip*/
if iSqrt(sum)**2 \== sum then iterate /*Not a perfect square? Then skip it. */
if iSqrt(dif)**2 \== dif then iterate /* " " " " " " " */
#= # + 1; call tell /*bump rare number counter; display #.*/
if #>=many then leave /* [↑] W: the width of # with commas.*/
end /*n*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg _; do jc=length(_)-3 to 1 by -3; _=insert(',', _, jc); end; return _ tell: say right(th(#),length(#)+9) ' rare number is:' right(commas(n),w); return th: parse arg th;return th||word('th st nd rd',1+(th//10)*(th//100%10\==1)*(th//10<4)) /*──────────────────────────────────────────────────────────────────────────────────────*/ iSqrt: parse arg x; $= 0; q= 1; do while q<=x; q=q*4; end
do while q>1; q=q%4; _= x-$-q; $= $%2; if _>=0 then do; x=_; $=$+q; end
end /*while q>1*/; return $</lang>
- output when using the input of: 8
1st rare number is: 65
2nd rare number is: 621,770
3rd rare number is: 281,089,082
4th rare number is: 2,022,652,202
5th rare number is: 2,042,832,002
6th rare number is: 868,591,084,757
7th rare number is: 872,546,974,178
8th rare number is: 872,568,754,178
Ring
<lang ring> load "stdlib.ring"
see "working..." + nl see "the first 5 rare numbers are:" + nl
num = 0
for n = 1 to 2042832002
strn = string(n)
nrev = ""
for m = len(strn) to 1 step -1
nrev = nrev + strn[m]
next
nrev = number(nrev)
sum = n + nrev
diff = n - nrev
if diff < 1
loop
ok
sqrtsum = sqrt(sum)
flagsum = (sqrtsum = floor(sqrtsum))
sqrtdiff = sqrt(diff)
flagdiff= (sqrtdiff = floor(sqrtdiff))
if flagsum = 1 and flagdiff = 1
num = num + 1
see "" + num + ": " + n + nl
ok
next see "done..." + nl </lang>
- Output:
working... the first 5 rare numbers are: 1: 65 2: 621770 3: 281089082 4: 2022652202 5: 2042832002 done...
Ruby
Not sure where, but there seems to be a bug that introduces false rare numbers as output beyond what is shown <lang ruby>Term = Struct.new(:coeff, :ix1, :ix2) do end
MAX_DIGITS = 16
def toLong(digits, reverse)
sum = 0
if reverse then
i = digits.length - 1
while i >=0
sum = sum *10 + digits[i]
i = i - 1
end
else
i = 0
while i < digits.length
sum = sum * 10 + digits[i]
i = i + 1
end
end
return sum
end
def isSquare(n)
root = Math.sqrt(n).to_i return root * root == n
end
def seq(from, to, step)
res = []
i = from
while i <= to
res << i
i = i + step
end
return res
end
def format_number(number)
number.to_s.reverse.gsub(/(\d{3})(?=\d)/, '\\1,').reverse
end
def main
pow = 1
allTerms = []
for i in 0 .. MAX_DIGITS - 2
allTerms << []
end
for r in 2 .. MAX_DIGITS
terms = []
pow = pow * 10
pow1 = pow
pow2 = 1
i1 = 0
i2 = r - 1
while i1 < i2
terms << Term.new(pow1 - pow2, i1, i2)
pow1 = (pow1 / 10).to_i
pow2 = pow2 * 10
i1 = i1 + 1
i2 = i2 - 1
end
allTerms[r - 2] = terms
end
# map of first minus last digits for 'n' to pairs giving this value
fml = {
0 =>[[2, 2], [8, 8]],
1 =>[[6, 5], [8, 7]],
4 =>4, 0,
6 =>[[6, 0], [8, 2]]
}
# map of other digit differences for 'n' to pairs giving this value
dmd = {}
for i in 0 .. 99
a = [(i / 10).to_i, (i % 10)]
d = a[0] - a[1]
if dmd.include?(d) then
dmd[d] << a
else
dmd[d] = [a]
end
end
fl = [0, 1, 4, 6]
dl = seq(-9, 9, 1) # all differences
zl = [0] # zero differences only
el = seq(-8, 8, 2) # even differences
ol = seq(-9, 9, 2) # odd differences only
il = seq(0, 9, 1)
rares = []
lists = []
for i in 0 .. 3
lists << []
end
fl.each_with_index { |f, i|
lists[i] = f
}
digits = []
count = 0
# Recursive closure to generate (n+r) candidates from (n-r) candidates
# and hence find Rare numbers with a given number of digits.
fnpr = lambda { |cand, di, dis, indices, nmr, nd, level|
if level == dis.length then
digits[indices[0][0]] = fml[cand[0]][di[0]][0]
digits[indices[0][1]] = fml[cand[0]][di[0]][1]
le = di.length
if nd % 2 == 1 then
le = le - 1
digits[(nd / 2).to_i] = di[le]
end
di[1 .. le - 1].each_with_index { |d, i|
digits[indices[i + 1][0]] = dmd[cand[i + 1]][d][0]
digits[indices[i + 1][1]] = dmd[cand[i + 1]][d][1]
}
r = toLong(digits, true)
npr = nmr + 2 * r
if not isSquare(npr) then
return
end
count = count + 1
print " R/N %2d:" % [count]
n = toLong(digits, false)
print " (%s)\n" % [format_number(n)]
rares << n
else
for num in dis[level]
di[level] = num
fnpr.call(cand, di, dis, indices, nmr, nd, level + 1)
end
end
}
# Recursive closure to generate (n-r) candidates with a given number of digits.
fnmr = lambda { |cand, list, indices, nd, level|
if level == list.length then
nmr = 0
nmr2 = 0
allTerms[nd - 2].each_with_index { |t, i|
if cand[i] >= 0 then
nmr = nmr + t.coeff * cand[i]
else
nmr2 = nmr2 = t.coeff * -cand[i]
if nmr >= nmr2 then
nmr = nmr - nmr2
nmr2 = 0
else
nmr2 = nmr2 - nmr
nmr = 0
end
end
}
if nmr2 >= nmr then
return
end
nmr = nmr - nmr2
if not isSquare(nmr) then
return
end
dis = []
dis << seq(0, fml[cand[0]].length - 1, 1)
for i in 1 .. cand.length - 1
dis << seq(0, dmd[cand[i]].length - 1, 1)
end
if nd % 2 == 1 then
dis << il.dup
end
di = []
for i in 0 .. dis.length - 1
di << 0
end
fnpr.call(cand, di, dis, indices, nmr, nd, 0)
else
for num in list[level]
cand[level] = num
fnmr.call(cand, list, indices, nd, level + 1)
end
end
}
#for nd in 2 .. MAX_DIGITS - 1
for nd in 2 .. 10
digits = []
for i in 0 .. nd - 1
digits << 0
end
if nd == 4 then
lists[0] << zl.dup
lists[1] << ol.dup
lists[2] << el.dup
lists[3] << ol.dup
elsif allTerms[nd - 2].length > lists[0].length then
for i in 0 .. 3
lists[i] << dl.dup
end
end
indices = []
for t in allTerms[nd - 2]
indices << [t.ix1, t.ix2]
end
for list in lists
cand = []
for i in 0 .. list.length - 1
cand << 0
end
fnmr.call(cand, list, indices, nd, 0)
end
print " %2d digits\n" % [nd]
end
rares.sort()
print "\nThe rare numbers with up to %d digits are:\n" % [MAX_DIGITS]
rares.each_with_index { |rare, i|
print " %2d: %25s\n" % [i + 1, format_number(rare)]
}
end
main()</lang>
- Output:
R/N 1: (65)
2 digits
3 digits
4 digits
5 digits
R/N 2: (621,770)
6 digits
7 digits
8 digits
R/N 3: (281,089,082)
9 digits
R/N 4: (2,022,652,202)
R/N 5: (2,042,832,002)
10 digits
The rare numbers with up to 16 digits are:
1: 65
2: 621,770
3: 281,089,082
4: 2,022,652,202
5: 2,042,832,002
Rust
A naive and an advanced version based on Nigel Galloway <lang rust> use itertools::Itertools; use std::collections::HashMap; use std::convert::TryInto; use std::fmt; use std::time::Instant;
- [derive(Debug)]
struct RareResults {
digits: u8, time_to_find: u128, counter: u32, number: u64,
}
impl fmt::Display for RareResults {
fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
write!(
f,
"{:>6} {:>6} ms {:>2}. {}",
self.digits, self.time_to_find, self.counter, self.number
)
}
}
fn print_results(results: Vec<RareResults>) {
if results.len() != 0 {
// println!("Results:");
println!("digits time #. Rare number");
for r in results {
println!("{}", r);
}
}
}
fn isqrt(n: u64) -> u64 {
let mut s = (n as f64).sqrt() as u64;
s = (s + n / s) >> 1;
if s * s > n {
s - 1
} else {
s
}
}
fn is_square(n: u64) -> bool {
match n & 0xf {
0 | 1 | 4 | 9 => {
let t = isqrt(n);
t * t == n
}
_ => false,
}
}
fn get_reverse(number: &u64) -> u64 {
number
.to_string()
.chars()
.map(|c| c.to_digit(10).unwrap())
.enumerate()
.fold(0_u64, |a, (i, d)| a + 10_u64.pow(i as u32) * d as u64)
} fn is_rare(number: u64) -> bool {
let reverse = get_reverse(&number);
reverse != number
&& number > reverse
&& is_square(number + reverse)
&& is_square(number - reverse)
}
/// This method is a very simple naive search, using brute-force to check a high amount of numbers /// for satisfying the rare number criterias. As such it is rather slow, and above 10 digits it's /// not really performant, release version takes ~30 secs to find the first 5 (max 10 digits) fn naive(digit: u8) -> Vec<RareResults> {
let bp_equal = (0_u8..=9).zip(0_u8..=9).collect::<Vec<(u8, u8)>>();
let bp_zero_or_even = (0_u8..=9)
.cartesian_product(0_u8..=9)
.filter(|pair| (pair.0 == pair.1) || (pair.0 as i32 - pair.1 as i32).abs() % 2 == 0)
.collect::<Vec<(u8, u8)>>();
let bp_odd = (0_u8..=9)
.cartesian_product(0_u8..=9)
.filter(|pair| (pair.0 as i32 - pair.1 as i32).abs() % 2 == 1)
.collect::<Vec<(u8, u8)>>();
let bp_9 = (0_u8..=9)
.cartesian_product(0_u8..=9)
.filter(|pair| pair.0 + pair.1 == 9)
.collect::<Vec<(u8, u8)>>();
let bp_73 = (0_u8..=9)
.cartesian_product(0_u8..=9)
.filter(|pair| [7, 3].contains(&(pair.0 as i8 - pair.1 as i8)))
.collect::<Vec<(u8, u8)>>();
let bp_11 = (0_u8..=9)
.cartesian_product(0_u8..=9)
.filter(|pair| pair.0 + pair.1 == 11 || pair.1 + pair.0 == 1)
.collect::<Vec<(u8, u8)>>();
let aq_bp_setup: Vec<((u8, u8), &Vec<(u8, u8)>)> = vec![
((2, 2), &bp_equal),
((4, 0), &bp_zero_or_even),
((6, 0), &bp_odd),
((6, 5), &bp_odd),
((8, 2), &bp_9),
((8, 3), &bp_73),
((8, 7), &bp_11),
((8, 8), &bp_equal),
];
//generate AB-PQ combinations
let aq_bp = aq_bp_setup
.iter()
.map(|e| {
e.1.iter().fold(vec![], |mut out, b| {
out.push(vec![e.0 .0, b.0, b.1, e.0 .1]);
out
})
})
.flatten()
.collect::<Vec<_>>();
let mut results: Vec<RareResults> = Vec::new(); let mut counter = 0_u32; let start_time = Instant::now();
let d = digit;
print!("Digits: {} ", d);
if d < 4 {
for n in 10_u64.pow((d - 1).into())..10_u64.pow(d.into()) {
if is_rare(n) {
counter += 1;
results.push(RareResults {
digits: d,
time_to_find: start_time.elapsed().as_millis(),
counter,
number: n,
});
}
}
} else {
aq_bp.iter().for_each(|abqp| {
let start = abqp[0] as u64 * 10_u64.pow((d - 1).into())
+ abqp[1] as u64 * 10_u64.pow((d - 2).into())
+ 10_u64 * abqp[2] as u64
+ abqp[3] as u64;
// brute-force checking all numbers which matches the pattern AB...PQ
// very slow
for n in (start..start + 10_u64.pow((d - 2).into())).step_by(100) {
if is_rare(n) {
counter += 1;
results.push(RareResults {
digits: d,
time_to_find: start_time.elapsed().as_millis(),
counter,
number: n,
});
}
}
});
}
println!(
"Digits: {} done - Elapsed time(ms): {}",
d,
start_time.elapsed().as_millis()
);
results
}
/// This algorithm uses an advanced search strategy based on Nigel Galloway's approach, /// and can find the first 40 rare numers (16 digits) within reasonable /// time in release version fn advanced(digit: u8) -> Vec<RareResults> {
// setup let mut results: Vec<RareResults> = Vec::new(); let mut counter = 0_u32; let start_time = Instant::now();
let numeric_digits = (0..=9).map(|x| [x, 0]).collect::<Vec<_>>(); let diffs1: Vec<i8> = vec![0, 1, 4, 5, 6];
// all possible digits pairs to calculate potential diffs
let pairs = (0_i8..=9)
.cartesian_product(0_i8..=9)
.map(|x| [x.0, x.1])
.collect::<Vec<_>>();
let all_diffs = (-9i8..=9).collect::<Vec<_>>();
// lookup table for the first diff
let lookup_1 = vec![
vec![[2, 2], [8, 8]], //Diff = 0
vec![[8, 7], [6, 5]], //Diff = 1
vec![],
vec![],
vec!4, 0, // Diff = 4
vec!8, 3, // Diff = 5
vec![[6, 0], [8, 2]], // Diff = 6
];
// lookup table for all the remaining diffs let lookup_n: HashMap<i8, Vec<_>> = pairs.into_iter().into_group_map_by(|elt| elt[0] - elt[1]);
let d = digit;
// powers like 1, 10, 100, 1000.... let powers = (0..d).map(|x| 10_u64.pow(x.into())).collect::<Vec<u64>>();
// for n-r (aka L) the required terms, like 9/ 99 / 999 & 90 / 99999 & 9999 & 900 etc
let terms = powers
.iter()
.zip(powers.iter().rev())
.map(|(a, b)| b.checked_sub(*a).unwrap_or(0))
.filter(|x| *x != 0)
.collect::<Vec<u64>>();
// create a cartesian product for all potential diff numbers
// for the first use the very short one, for all other the complete 19 element
let diff_list_iter = (0_u8..(d / 2))
.map(|i| match i {
0 => diffs1.iter(),
_ => all_diffs.iter(),
})
.multi_cartesian_product()
// remove invalid first diff/second diff combinations - custom iterator would be probably better
.filter(|x| {
if x.len() == 1 {
return true;
}
match (*x[0], *x[1]) {
(a, b) if (a == 0 && b != 0) => false,
(a, b) if (a == 1 && ![-7, -5, -3, -1, 1, 3, 5, 7].contains(&b)) => false,
(a, b) if (a == 4 && ![-8, -6, -4, -2, 0, 2, 4, 6, 8].contains(&b)) => false,
(a, b) if (a == 5 && ![7, -3].contains(&b)) => false,
(a, b) if (a == 6 && ![-9, -7, -5, -3, -1, 1, 3, 5, 7, 9].contains(&b)) => {
false
}
_ => true,
}
});
#[cfg(debug_assertions)]
{
println!(" powers: {:?}", powers);
println!(" terms: {:?}", terms);
}
diff_list_iter.for_each(|diffs| {
// calculate difference of original n and its reverse (aka L = n-r)
// which must be a perfect square
let l: i64 = diffs
.iter()
.zip(terms.iter())
.map(|(diff, term)| **diff as i64 * *term as i64)
.sum();
if l > 0 && is_square(l.try_into().unwrap()) {
// potential candiate, at least L is a perfect square
#[cfg(debug_assertions)]
println!(" square L: {}, diffs: {:?}", l, diffs);
// placeholder for the digits
let mut dig: Vec<i8> = vec![0_i8; d.into()];
// generate a cartesian product for each identified diff using the lookup tables
let c_iter = (0..(diffs.len() + d as usize % 2))
.map(|i| match i {
0 => lookup_1[*diffs[0] as usize].iter(),
_ if i != diffs.len() => lookup_n.get(diffs[i]).unwrap().iter(),
_ => numeric_digits.iter(), // for the middle digits
})
.multi_cartesian_product();
// check each H (n+r) by using digit combination
c_iter.for_each(|elt| {
// print!(" digits combinations: {:?}", elt);
for (i, digit_pair) in elt.iter().enumerate() {
// print!(" digit pairs: {:?}, len: {}", digit_pair, l.len());
dig[i] = digit_pair[0];
dig[d as usize - 1 - i] = digit_pair[1]
}
// for numbers with odd # digits restore the middle digit
// which has been overwritten at the end of the previous cycle
if d % 2 == 1 {
dig[(d as usize - 1) / 2] = elt[elt.len() - 1][0];
}
let num = dig
.iter()
.rev()
.enumerate()
.fold(0_u64, |acc, (i, d)| acc + 10_u64.pow(i as u32) * *d as u64);
let reverse = dig
.iter()
.enumerate()
.fold(0_u64, |acc, (i, d)| acc + 10_u64.pow(i as u32) * *d as u64);
if num > reverse && is_square(num + reverse) {
println!(" FOUND: {}, reverse: {}", num, reverse);
counter += 1;
results.push(RareResults {
digits: d,
time_to_find: start_time.elapsed().as_millis(),
counter,
number: num,
});
}
});
}
});
println!(
"Digits: {} done - Elapsed time(ms): {}",
d,
start_time.elapsed().as_millis()
);
results
} fn main() {
println!("Run this program in release mode for measuring performance");
println!("Naive version:");
(1..=10).for_each(|x| print_results(naive(x)));
println!("Advanced version:");
(1..=15).for_each(|x| print_results(advanced(x)));
}
</lang>
- Output:
Run this program in release mode for measuring performance
Naive version:
Digits: 1 Digits: 1 done - Elapsed time(ms): 0
Digits: 2 Digits: 2 done - Elapsed time(ms): 0
digits time #. Rare number
2 0 ms 1. 65
Digits: 3 Digits: 3 done - Elapsed time(ms): 0
Digits: 4 Digits: 4 done - Elapsed time(ms): 0
Digits: 5 Digits: 5 done - Elapsed time(ms): 0
Digits: 6 Digits: 6 done - Elapsed time(ms): 11
digits time #. Rare number
6 3 ms 1. 621770
Digits: 7 Digits: 7 done - Elapsed time(ms): 76
Digits: 8 Digits: 8 done - Elapsed time(ms): 641
Digits: 9 Digits: 9 done - Elapsed time(ms): 6375
digits time #. Rare number
9 232 ms 1. 281089082
Digits: 10 Digits: 10 done - Elapsed time(ms): 37598
digits time #. Rare number
10 44 ms 1. 2022652202
10 82 ms 2. 2042832002
Advanced version:
Digits: 1 done - Elapsed time(ms): 6
FOUND: 65, reverse: 56
Digits: 2 done - Elapsed time(ms): 0
digits time #. Rare number
2 0 ms 1. 65
Digits: 3 done - Elapsed time(ms): 0
Digits: 4 done - Elapsed time(ms): 0
Digits: 5 done - Elapsed time(ms): 0
FOUND: 621770, reverse: 77126
Digits: 6 done - Elapsed time(ms): 0
digits time #. Rare number
6 0 ms 1. 621770
Digits: 7 done - Elapsed time(ms): 0
Digits: 8 done - Elapsed time(ms): 5
FOUND: 281089082, reverse: 280980182
Digits: 9 done - Elapsed time(ms): 5
digits time #. Rare number
9 0 ms 1. 281089082
FOUND: 2022652202, reverse: 2022562202
FOUND: 2042832002, reverse: 2002382402
Digits: 10 done - Elapsed time(ms): 107
digits time #. Rare number
10 6 ms 1. 2022652202
10 22 ms 2. 2042832002
Digits: 11 done - Elapsed time(ms): 161
FOUND: 872546974178, reverse: 871479645278
FOUND: 872568754178, reverse: 871457865278
FOUND: 868591084757, reverse: 757480195868
Digits: 12 done - Elapsed time(ms): 1707
digits time #. Rare number
12 319 ms 1. 872546974178
12 342 ms 2. 872568754178
12 846 ms 3. 868591084757
FOUND: 6979302951885, reverse: 5881592039796
Digits: 13 done - Elapsed time(ms): 2409
digits time #. Rare number
13 540 ms 1. 6979302951885
FOUND: 20313693904202, reverse: 20240939631302
FOUND: 20313839704202, reverse: 20240793831302
FOUND: 20331657922202, reverse: 20222975613302
FOUND: 20331875722202, reverse: 20222757813302
FOUND: 20333875702202, reverse: 20220757833302
FOUND: 40313893704200, reverse: 240739831304
FOUND: 40351893720200, reverse: 202739815304
Digits: 14 done - Elapsed time(ms): 41440
digits time #. Rare number
14 6568 ms 1. 20313693904202
14 6642 ms 2. 20313839704202
14 7727 ms 3. 20331657922202
14 7881 ms 4. 20331875722202
14 8287 ms 5. 20333875702202
14 25737 ms 6. 40313893704200
14 25845 ms 7. 40351893720200
FOUND: 200142385731002, reverse: 200137583241002
FOUND: 221462345754122, reverse: 221457543264122
FOUND: 816984566129618, reverse: 816921665489618
FOUND: 245518996076442, reverse: 244670699815542
FOUND: 204238494066002, reverse: 200660494832402
FOUND: 248359494187442, reverse: 244781494953842
FOUND: 244062891224042, reverse: 240422198260442
FOUND: 403058392434500, reverse: 5434293850304
FOUND: 441054594034340, reverse: 43430495450144
Digits: 15 done - Elapsed time(ms): 42591
digits time #. Rare number
15 2788 ms 1. 200142385731002
15 2970 ms 2. 221462345754122
15 5616 ms 3. 816984566129618
15 6999 ms 4. 245518996076442
15 7229 ms 5. 204238494066002
15 7291 ms 6. 248359494187442
15 7547 ms 7. 244062891224042
15 23704 ms 8. 403058392434500
15 23883 ms 9. 441054594034340
Visual Basic .NET
Traditional
via
Surprisingly slow, I expected performance to be a little slower than C#, but this is quite a bit slower. This vb.net version takes 1 2/3 minutes to do what the C# version can do in 2/3 of a minute.
<lang vbnet>Imports System.Console Imports DT = System.DateTime Imports Lsb = System.Collections.Generic.List(Of SByte) Imports Lst = System.Collections.Generic.List(Of System.Collections.Generic.List(Of SByte)) Imports UI = System.UInt64
Module Module1
Const MxD As SByte = 15
Public Structure term
Public coeff As UI : Public a, b As SByte
Public Sub New(ByVal c As UI, ByVal a_ As Integer, ByVal b_ As Integer)
coeff = c : a = CSByte(a_) : b = CSByte(b_)
End Sub
End Structure
Dim nd, nd2, count As Integer, digs, cnd, di As Integer() Dim res As List(Of UI), st As DT, tLst As List(Of List(Of term)) Dim lists As List(Of Lst), fml, dmd As Dictionary(Of Integer, Lst) Dim dl, zl, el, ol, il As Lsb, odd As Boolean, ixs, dis As Lst, Dif As UI
' converts digs array to the "difference"
Function ToDif() As UI
Dim r As UI = 0 : For i As Integer = 0 To digs.Length - 1 : r = r * 10 + digs(i)
Next : Return r
End Function
' converts digs array to the "sum"
Function ToSum() As UI
Dim r As UI = 0 : For i As Integer = digs.Length - 1 To 0 Step -1 : r = r * 10 + digs(i)
Next : Return Dif + (r << 1)
End Function
' determines if the nmbr is square or not
Function IsSquare(nmbr As UI) As Boolean
If (&H202021202030213 And (1UL << (nmbr And 63))) <> 0 Then _
Dim r As UI = Math.Sqrt(nmbr) : Return r * r = nmbr Else Return False
End Function
'// returns sequence of SBbytes
Function Seq(from As SByte, upto As Integer, Optional stp As SByte = 1) As Lsb
Dim res As Lsb = New Lsb()
For item As SByte = from To upto Step stp : res.Add(item) : Next : Return res
End Function
' Recursive closure to generate (n+r) candidates from (n-r) candidates
Sub Fnpr(ByVal lev As Integer)
If lev = dis.Count Then
digs(ixs(0)(0)) = fml(cnd(0))(di(0))(0) : digs(ixs(0)(1)) = fml(cnd(0))(di(0))(1)
Dim le As Integer = di.Length, i As Integer = 1
If odd Then le -= 1 : digs(nd >> 1) = di(le)
For Each d As SByte In di.Skip(1).Take(le - 1)
digs(ixs(i)(0)) = dmd(cnd(i))(d)(0)
digs(ixs(i)(1)) = dmd(cnd(i))(d)(1) : i += 1 : Next
If Not IsSquare(ToSum()) Then Return
res.Add(ToDif()) : count += 1
WriteLine("{0,16:n0}{1,4} ({2:n0})", (DT.Now - st).TotalMilliseconds, count, res.Last())
Else
For Each n In dis(lev) : di(lev) = n : Fnpr(lev + 1) : Next
End If
End Sub
' Recursive closure to generate (n-r) candidates with a given number of digits.
Sub Fnmr(ByVal list As Lst, ByVal lev As Integer)
If lev = list.Count Then
Dif = 0 : Dim i As SByte = 0 : For Each t In tLst(nd2)
If cnd(i) < 0 Then Dif -= t.coeff * CULng(-cnd(i)) _
Else Dif += t.coeff * CULng(cnd(i))
i += 1 : Next
If Dif <= 0 OrElse Not IsSquare(Dif) Then Return
dis = New Lst From {Seq(0, fml(cnd(0)).Count - 1)}
For Each i In cnd.Skip(1) : dis.Add(Seq(0, dmd(i).Count - 1)) : Next
If odd Then dis.Add(il)
di = New Integer(dis.Count - 1) {} : Fnpr(0)
Else
For Each n As SByte In list(lev) : cnd(lev) = n : Fnmr(list, lev + 1) : Next
End If
End Sub
Sub init()
Dim pow As UI = 1
' terms of (n-r) expression for number of digits from 2 to maxDigits
tLst = New List(Of List(Of term))() : For Each r As Integer In Seq(2, MxD)
Dim terms As List(Of term) = New List(Of term)()
pow *= 10 : Dim p1 As UI = pow, p2 As UI = 1
Dim i1 As Integer = 0, i2 As Integer = r - 1
While i1 < i2 : terms.Add(New term(p1 - p2, i1, i2))
p1 = p1 / 10 : p2 = p2 * 10 : i1 += 1 : i2 -= 1 : End While
tLst.Add(terms) : Next
' map of first minus last digits for 'n' to pairs giving this value
fml = New Dictionary(Of Integer, Lst)() From {
{0, New Lst() From {New Lsb() From {2, 2}, New Lsb() From {8, 8}}},
{1, New Lst() From {New Lsb() From {6, 5}, New Lsb() From {8, 7}}},
{4, New Lst() From {New Lsb() From {4, 0}}},
{6, New Lst() From {New Lsb() From {6, 0}, New Lsb() From {8, 2}}}}
' map of other digit differences for 'n' to pairs giving this value
dmd = New Dictionary(Of Integer, Lst)()
For i As SByte = 0 To 10 - 1 : Dim j As SByte = 0, d As SByte = i
While j < 10 : If dmd.ContainsKey(d) Then dmd(d).Add(New Lsb From {i, j}) _
Else dmd(d) = New Lst From {New Lsb From {i, j}}
j += 1 : d -= 1 : End While : Next
dl = Seq(-9, 9) ' all differences
zl = Seq(0, 0) ' zero difference
el = Seq(-8, 8, 2) ' even differences
ol = Seq(-9, 9, 2) ' odd differences
il = Seq(0, 9)
lists = New List(Of Lst)()
For Each f As SByte In fml.Keys : lists.Add(New Lst From {New Lsb From {f}}) : Next
End Sub
Sub Main(ByVal args As String())
init() : res = New List(Of UI)() : st = DT.Now : count = 0
WriteLine("{0,5}{1,12}{2,4}{3,14}", "digs", "elapsed(ms)", "R/N", "Rare Numbers")
nd = 2 : nd2 = 0 : odd = False : While nd <= MxD
digs = New Integer(nd - 1) {} : If nd = 4 Then
lists(0).Add(zl) : lists(1).Add(ol) : lists(2).Add(el) : lists(3).Add(ol)
ElseIf tLst(nd2).Count > lists(0).Count Then
For Each list As Lst In lists : list.Add(dl) : Next : End If
ixs = New Lst() : For Each t As term In tLst(nd2) : ixs.Add(New Lsb From {t.a, t.b}) : Next
For Each list As Lst In lists : cnd = New Integer(list.Count - 1) {} : Fnmr(list, 0) : Next
WriteLine(" {0,2} {1,10:n0}", nd, (DT.Now - st).TotalMilliseconds)
nd += 1 : nd2 += 1 : odd = Not odd : End While
res.Sort() : WriteLine(vbLf & "The {0} rare numbers with up to {1} digits are:", res.Count, MxD)
count = 0 : For Each rare In res : count += 1 : WriteLine("{0,2}:{1,27:n0}", count, rare) : Next
If System.Diagnostics.Debugger.IsAttached Then ReadKey()
End Sub
End Module</lang>
- Output:
digs elapsed(ms) R/N Rare Numbers
25 1 (65)
2 26
3 26
4 27
5 27
28 2 (621,770)
6 29
7 30
8 41
42 3 (281,089,082)
9 46
47 4 (2,022,652,202)
116 5 (2,042,832,002)
10 273
11 422
1,363 6 (872,546,974,178)
1,476 7 (872,568,754,178)
2,937 8 (868,591,084,757)
12 3,584
4,560 9 (6,979,302,951,885)
13 5,817
18,234 10 (20,313,693,904,202)
18,471 11 (20,313,839,704,202)
23,626 12 (20,331,657,922,202)
24,454 13 (20,331,875,722,202)
26,599 14 (20,333,875,702,202)
60,784 15 (40,313,893,704,200)
61,246 16 (40,351,893,720,200)
14 65,387
65,465 17 (200,142,385,731,002)
66,225 18 (221,462,345,754,122)
76,417 19 (816,984,566,129,618)
81,727 20 (245,518,996,076,442)
82,461 21 (204,238,494,066,002)
82,694 22 (248,359,494,187,442)
83,729 23 (244,062,891,224,042)
99,241 24 (403,058,392,434,500)
100,009 25 (441,054,594,034,340)
15 104,207
The 25 rare numbers with up to 15 digits are:
1: 65
2: 621,770
3: 281,089,082
4: 2,022,652,202
5: 2,042,832,002
6: 868,591,084,757
7: 872,546,974,178
8: 872,568,754,178
9: 6,979,302,951,885
10: 20,313,693,904,202
11: 20,313,839,704,202
12: 20,331,657,922,202
13: 20,331,875,722,202
14: 20,333,875,702,202
15: 40,313,893,704,200
16: 40,351,893,720,200
17: 200,142,385,731,002
18: 204,238,494,066,002
19: 221,462,345,754,122
20: 244,062,891,224,042
21: 245,518,996,076,442
22: 248,359,494,187,442
23: 403,058,392,434,500
24: 441,054,594,034,340
25: 816,984,566,129,618
Quicker
(translation of the quicker version)
Performance is better, only about 4% slower than C#. <lang vbnet>Imports System.Math Imports System.Console Imports llst = System.Collections.Generic.List(Of Integer())
Module Module1
Dim d, dac As Integer(), drar As Integer() = New Integer(19) {} : Dim ac, pp As Long(), p As Long() = New Long(18) {}
Dim odd As Boolean = False : Dim sum, rt As Long : Dim ln, dl As Integer, cn As Integer = 0, nd As Integer = 2, nd1 As Integer = nd - 1
Dim sw As Stopwatch = New Stopwatch(), swt As Stopwatch = New Stopwatch() : Dim sr As List(Of Long) = New List(Of Long)()
ReadOnly tlo As Integer() = New Integer() {0, 1, 4, 5, 6}, all As Integer() = Seq(-9, 9), odl As Integer() = Seq(-9, 9, 2), evl As Integer() = Seq(-8, 8, 2),
thi As Integer() = New Integer() {4, 5, 6, 9, 10, 11, 14, 15, 16}, alh As Integer() = Seq(0, 18), odh As Integer() = Seq(1, 17, 2),
evh As Integer() = Seq(0, 18, 2), ten As Integer() = Seq(0, 9), z As Integer() = Seq(0, 0), t7 As Integer() = New Integer() {-3, 7}, nin As Integer() = New Integer() {9}, tn As Integer() = New Integer() {10}, t12 As Integer() = New Integer() {2, 12}, o11 As Integer() = New Integer() {1, 11}, pos As Integer() = New Integer() {0, 1, 4, 5, 6, 9}
Dim lu, l2 As llst, lul As llst = New llst From {z, odl, Nothing, Nothing, evl, t7, odl},
luh As llst = New llst From {tn, evh, Nothing, Nothing, evh, t12, odh, Nothing, Nothing, evh, nin, odh, Nothing, Nothing, odh, o11, evh},
l2l As llst = New llst From {pos, Nothing, Nothing, Nothing, all, Nothing, all},
l2h As llst = New llst From {Nothing, Nothing, Nothing, Nothing, alh, Nothing, alh, Nothing, Nothing, Nothing, alh, Nothing, Nothing, Nothing, alh, Nothing, alh}
Dim chTen As Integer()() = New Integer()() {New Integer() {0, 2, 5, 8, 9}, New Integer() {0, 3, 4, 6, 9}, New Integer() {1, 4, 7, 8},
New Integer() {2, 3, 5, 8}, New Integer() {0, 3, 6, 7, 9}, New Integer() {1, 2, 4, 7},
New Integer() {2, 5, 6, 8}, New Integer() {0, 1, 3, 6, 9}, New Integer() {1, 4, 5, 7}}
Dim chAH As Integer()() = New Integer()() {
New Integer() {0, 2, 5, 8, 9, 11, 14, 17, 18}, New Integer() {0, 3, 4, 6, 9, 12, 13, 15, 18}, New Integer() {1, 4, 7, 8, 10, 13, 16, 17},
New Integer() {2, 3, 5, 8, 11, 12, 14, 17}, New Integer() {0, 3, 6, 7, 9, 12, 15, 16, 18}, New Integer() {1, 2, 4, 7, 10, 11, 13, 16},
New Integer() {2, 5, 6, 8, 11, 14, 15, 17}, New Integer() {0, 1, 3, 6, 9, 10, 12, 15, 18}, New Integer() {1, 4, 5, 7, 10, 13, 14, 16}}
Function Seq(ByVal f As Integer, ByVal t As Integer, ByVal Optional s As Integer = 1) As Integer()
Dim r As Integer() = New Integer((t - f) / s + 1 - 1) {}
For i As Integer = 0 To r.Length - 1 : r(i) = f : f += s : Next : Return r : End Function
Function ISR(ByVal s As Long) As Long
Return Sqrt(s) : End Function
Function IsRev(ByVal nd As Integer, ByVal f As Long, ByVal r As Long) As Boolean
nd -= 1 : Return If(f \ p(nd) <> r Mod 10, False, (If(nd < 1, True, IsRev(nd, f Mod p(nd), r \ 10)))) : End Function
Sub RecurseLE5(ByVal lst As llst, ByVal lv As Integer)
If lv = dl Then
sum = ac(lv - 1) : If sum > 0 Then rt = CLng(Sqrt(sum)) : If rt * rt = sum Then sr.Add(sum)
Else For Each n As Integer In lst(lv)
d(lv) = n : If lv = 0 Then ac(0) = pp(0) * n Else ac(lv) = ac(lv - 1) + pp(lv) * n
RecurseLE5(lst, lv + 1) : Next : End If : End Sub
Sub Recursehi(ByVal lst As llst, ByVal lv As Integer)
Dim lv1 As Integer = lv - 1 : If lv = dl Then
sum = ac(lv1) : If (&H202021202030213 And (1L << (sum And 63))) > 0 Then rt = CLng(Sqrt(sum)) : If rt * rt = sum Then sr.Add(sum)
Else For Each n As Integer In lst(lv)
d(lv) = n : If lv = 0 Then ac(0) = pp(0) * n : dac(0) = drar(n) _
Else ac(lv) = ac(lv1) + pp(lv) * n : dac(lv) = dac(lv1) + drar(n) : If dac(lv) > 8 Then dac(lv) -= 9
Select Case lv
Case 0 : ln = n : lst(1) = lu(n) : lst(2) = l2(n)
Case 1 : Select Case ln
Case 5, 15 : lst(2) = If(n < 10, evh, odh)
Case 9 : lst(2) = If(((n >> 1) And 1) = 0, evh, odh)
Case 11 : lst(2) = If(((n >> 1) And 1) = 1, evh, odh)
End Select : End Select
If lv = dl - 2 Then lst(dl - 1) = If(odd, chTen(dac(dl - 2)), chAH(dac(dl - 2)))
Recursehi(lst, lv + 1) : Next : End If : End Sub
Sub Recurselo(ByVal lst As llst, ByVal lv As Integer)
Dim lv1 As Integer = lv - 1 : If lv = dl Then
sum = ac(lv1) : If sum > 0 Then rt = CLng(Sqrt(sum)) : If rt * rt = sum Then sr.Add(sum)
Else For Each n As Integer In lst(lv)
d(lv) = n : If lv = 0 Then ac(0) = pp(0) * n Else ac(lv) = ac(lv1) + pp(lv) * n
Select Case lv
Case 0 : ln = n : lst(1) = lu(n) : lst(2) = l2(n)
Case 1 : Select Case ln
Case 1 : lst(2) = If((((n + 9) >> 1) And 1) = 0, evl, odl)
Case 5 : lst(2) = If(n < 0, evl, odl)
End Select : End Select
Recurselo(lst, lv + 1) : Next : End If : End Sub
Function listEm(ByVal lst As llst, ByVal plu As llst, ByVal pl2 As llst) As List(Of Long)
dl = lst.Count : d = New Integer(dl - 1) {} : sr.Clear() : lu = plu : l2 = pl2
ac = New Long(dl - 1) {} : dac = New Integer(dl - 1) {} : pp = New Long(dl - 1) {}
Dim j As Integer = nd1 : For i As Integer = 0 To dl - 1 : pp(i) = If(lst(0).Length > 6, p(j) + p(i), p(j) - p(i)) : j -= 1 : Next
If nd <= 5 Then RecurseLE5(lst, 0) Else If lst(0).Length > 6 Then Recursehi(lst, 0) Else Recurselo(lst, 0)
Return sr : End Function
Sub Reveal(ByVal lo As List(Of Long), ByVal hi As List(Of Long))
Dim s As List(Of String) = New List(Of String)() : For Each l As Long In lo : For Each h As Long In hi
Dim r As Long = (h - l) \ 2, f As Long = h - r
If IsRev(nd, f, r) Then s.Add(String.Format("{0,20} {1,11} {2,10} ", f, ISR(h), ISR(l)))
Next : Next : s.Sort() : If s.Count > 0 Then _
For Each t As String In s : cn += 1 : Write("{0,2} {1}{2}", cn, t, If(t = s.Last(), "", vbLf)) : Next Else Write("{0,48}", "")
End Sub
Sub Main(ByVal args As String())
WriteLine("{0,3}{1,20} {2,11} {3,10} {4,4}{5,16} {6, 17}", "nth", "forward", "rt.sum", "rt.dif", "digs", "block time", "total time")
p(0) = 1 : Dim j As Integer = 0 : For i As Integer = 1 To p.Length - 1 : p(i) = p(j) * 10 : j = i : Next
For i As Integer = 0 To drar.Length - 1 : drar(i) = (i * 2) Mod 9 : Next
Dim lls As llst = New llst From {tlo}, hls As llst = New llst From {thi} : sw.Start() : swt.Start()
While nd <= 18
If nd > 2 Then If odd Then hls.Add(ten) Else lls.Add(all) : hls(hls.Count - 1) = alh
Reveal(listEm(lls, lul, l2l).ToList(), listEm(hls, luh, l2h))
If Not odd AndAlso nd > 5 Then hls(hls.Count - 1) = alh
WriteLine("{0,2}: {1} {2}", nd, sw.Elapsed, swt.Elapsed) : sw.Restart()
nd1 = nd : nd += 1 : odd = Not odd
End While
' 19
hls.Add(ten)
Reveal(listEmU(lls, lul, l2l).ToList(), listEmU(hls, luh, l2h))
WriteLine("{0,2}: {1} {2}", nd, sw.Elapsed, swt.Elapsed) : End Sub
- Region "19"
Dim usum, urt As ULong Dim acu, ppu As ULong() Dim sru As List(Of ULong) = New List(Of ULong)()
Sub Reveal(ByVal lo As List(Of ULong), ByVal hi As List(Of ULong))
Dim s As List(Of String) = New List(Of String)() : For Each l As ULong In lo : For Each h As ULong In hi
Dim r As ULong = (h - l) >> 1, f As ULong = h - r
If IsRev(nd, f, r) Then s.Add(String.Format("{0,20} {1,11} {2,10} ", f, ISR(h), ISR(l)))
Next : Next : s.Sort() : If s.Count > 0 Then _
For Each t As String In s : cn += 1 : Write("{0,2} {1}{2}", cn, t, If(t = s.Last(), "", vbLf)) : Next Else Write("{0,48}", "")
End Sub
Function listEmU(ByVal lst As llst, ByVal plu As llst, ByVal pl2 As llst) As List(Of ULong)
dl = lst.Count : d = New Integer(dl - 1) {} : sru.Clear() : lu = plu : l2 = pl2
acu = New ULong(dl - 1) {} : dac = New Integer(dl - 1) {} : ppu = New ULong(dl - 1) {}
Dim j As Integer = nd1 : For i As Integer = 0 To dl - 1 : ppu(i) = CULng(If(lst(0).Length > 6, p(j) + p(i), p(j) - p(i))) : j -= 1 : Next
If lst(0).Length > 8 Then RecurseUhi(lst, 0) Else RecurseUlo(lst, 0)
Return sru : End Function
Sub RecurseUhi(ByVal lst As llst, ByVal lv As Integer)
Dim lv1 As Integer = lv - 1 : If lv = dl Then
usum = acu(lv1)
If (&H202021202030213 And (1UL << (usum And 63))) <> 0 Then urt = Sqrt(usum) : If urt * urt = usum Then sru.Add(usum)
Else For Each n As Integer In lst(lv)
d(lv) = n : If lv = 0 Then
acu(0) = ppu(0) * CUInt(n) : dac(0) = drar(n)
Else
acu(lv) = If(n >= 0, acu(lv1) + ppu(lv) * CUInt(n), acu(lv1) - ppu(lv) * CUInt(-n))
dac(lv) = dac(lv1) + drar(n) : If dac(lv) > 8 Then dac(lv) -= 9
End If
Select Case lv
Case 0 : ln = n : lst(1) = lu(n) : lst(2) = l2(n)
Case 1 : Select Case ln
Case 5, 15 : lst(2) = If(n < 10, evh, odh)
Case 9 : lst(2) = If(((n >> 1) And 1) = 0, evh, odh)
Case 11 : lst(2) = If(((n >> 1) And 1) = 1, evh, odh)
End Select : End Select
If lv = dl - 2 Then lst(dl - 1) = If(odd, chTen(dac(dl - 2)), chAH(dac(dl - 2)))
RecurseUhi(lst, lv + 1) : Next : End If : End Sub
Sub RecurseUlo(ByVal lst As llst, ByVal lv As Integer)
Dim lv1 As Integer = lv - 1 : If lv = dl Then
usum = acu(lv1)
If usum > 0 Then urt = Sqrt(usum) : If urt * urt = usum Then sru.Add(usum)
Else For Each n As Integer In lst(lv)
d(lv) = n : If lv = 0 Then acu(0) = ppu(0) * CUInt(n) Else _
acu(lv) = If(n >= 0, acu(lv1) + ppu(lv) * CUInt(n), acu(lv1) - ppu(lv) * CUInt(-n))
Select Case lv
Case 0 : ln = n : lst(1) = lu(n) : lst(2) = l2(n)
Case 1 : Select Case ln
Case 1 : lst(2) = If((((n + 9) >> 1) And 1) = 0, evl, odl)
Case 5 : lst(2) = If(n < 0, evl, odl)
End Select : End Select
RecurseUlo(lst, lv + 1) : Next : End If : End Sub
Function ISR(ByVal s As ULong) As ULong
Return Sqrt(s) : End Function
Function IsRev(ByVal nd As Integer, ByVal f As ULong, ByVal r As ULong) As Boolean
nd -= 1 : Return If(f \ CULng(p(nd)) <> r Mod 10, False, (If(nd < 1, True, IsRev(nd, f Mod CULng(p(nd)), r \ 10UL)))) : End Function
- End Region
End Module</lang> Results on the core i7-7700 @ 3.6Ghz.
nth forward rt.sum rt.dif digs block time total time
1 65 11 3 2: 00:00:00.0037657 00:00:00.0037657
3: 00:00:00.0001034 00:00:00.0040327
4: 00:00:00.0000951 00:00:00.0042102
5: 00:00:00.0000928 00:00:00.0043867
2 621770 836 738 6: 00:00:00.0022229 00:00:00.0066922
7: 00:00:00.0001703 00:00:00.0069250
8: 00:00:00.0002647 00:00:00.0072713
3 281089082 23708 330 9: 00:00:00.0013320 00:00:00.0086847
4 2022652202 63602 300
5 2042832002 63602 6360 10: 00:00:00.0043004 00:00:00.0130515
11: 00:00:00.0202717 00:00:00.0334094
6 868591084757 1275175 333333
7 872546974178 1320616 32670
8 872568754178 1320616 33330 12: 00:00:00.0553298 00:00:00.0889160
9 6979302951885 3586209 1047717 13: 00:00:00.3615348 00:00:00.4505467
10 20313693904202 6368252 269730
11 20313839704202 6368252 270270
12 20331657922202 6368252 329670
13 20331875722202 6368252 330330
14 20333875702202 6368252 336330
15 40313893704200 6368252 6330336
16 40351893720200 6368252 6336336 14: 00:00:00.9808061 00:00:01.4315251
17 200142385731002 20006998 69300
18 204238494066002 20122102 1891560
19 221462345754122 21045662 69300
20 244062891224042 22011022 1908060
21 245518996076442 22140228 921030
22 248359494187442 22206778 1891560
23 403058392434500 20211202 19940514
24 441054594034340 22011022 19940514
25 816984566129618 40421606 250800 15: 00:00:06.8062687 00:00:08.2378833
26 2078311262161202 64030648 7529850
27 2133786945766212 65272218 2666730
28 2135568943984212 65272218 3267330
29 2135764587964212 65272218 3326670
30 2135786765764212 65272218 3333330
31 4135786945764210 65272218 63333336
32 6157577986646405 105849161 33333333
33 6889765708183410 83866464 82133718
34 8052956026592517 123312255 29999997
35 8052956206592517 123312255 30000003
36 8191154686620818 127950856 3299670
37 8191156864620818 127950856 3300330
38 8191376864400818 127950856 3366330
39 8650327689541457 127246955 33299667
40 8650349867341457 127246955 33300333 16: 00:00:18.5333654 00:00:26.7713563
41 22542040692914522 212329862 333300
42 67725910561765640 269040196 251135808
43 86965750494756968 417050956 33000 17: 00:02:08.5301411 00:02:35.3016150
44 225342456863243522 671330638 297000
45 225342458663243522 671330638 303000
46 225342478643243522 671330638 363000
47 284684666566486482 754565658 30000
48 284684868364486482 754565658 636000
49 297128548234950692 770186978 32697330
50 297128722852950692 770186978 32702670
51 297148324656930692 770186978 33296670
52 297148546434930692 770186978 33303330
53 497168548234910690 770186978 633363336
54 619431353040136925 1071943279 299667003
55 619631153042134925 1071943279 300333003
56 631688638047992345 1083968809 297302703
57 633288858025996145 1083968809 302637303
58 633488632647994145 1083968809 303296697
59 653488856225994125 1083968809 363303363
60 811865096390477018 1273828556 33030330
61 865721270017296468 1315452006 32071170
62 871975098681469178 1320582934 3303300
63 898907259301737498 1339270086 64576740 18: 00:05:50.9239434 00:08:26.2256326
64 2042401829204402402 2021001202 18915600
65 2060303819041450202 2020110202 199405140
66 2420424089100600242 2200110022 19080600
67 2551755006254571552 2259094848 693000
68 2702373360882732072 2324811012 693000
69 2825378427312735282 2377130742 2508000
70 6531727101458000045 3454234451 1063822617
71 6988066446726832640 2729551744 2554541088
72 8066308349502036608 4016542096 2508000
73 8197906905009010818 4046976144 133408770
74 8200756128308135597 4019461925 495417087
75 8320411466598809138 4079154376 36366330 19: 00:45:09.0009635 00:53:36.4768204
Wren
A translation of Go's 'traditional' and 'turbo' versions.
Quite a tough task for the little bird which lacks the speed of its compiled, statically typed brethren. Also integer arithmetic is unreliable for numbers >= 2^53 which effectively limits us here to 15 digit rare numbers without resorting to BigInt which would be even slower.
Traditional
About 9.5 minutes to find the first 25 rare numbers. <lang ecmascript>import "/sort" for Sort import "/fmt" for Fmt
class Term {
construct new(coeff, ix1, ix2) {
_coeff = coeff
_ix1 = ix1
_ix2 = ix2
}
coeff { _coeff }
ix1 { _ix1 }
ix2 { _ix2 }
}
var maxDigits = 15
var toInt = Fn.new { |digits, reverse|
var sum = 0
if (!reverse) {
for (i in 0...digits.count) sum = sum*10 + digits[i]
} else {
for (i in digits.count-1..0) sum = sum*10 + digits[i]
}
return sum
}
var isSquare = Fn.new { |n|
var root = n.sqrt.floor return root*root == n
}
var seq = Fn.new { |from, to, step|
var res = []
var i = from
while (i <= to) {
res.add(i)
i = i + step
}
return res
}
var start = System.clock var pow = 1 System.print("Aggregate timings to process all numbers up to:")
// terms of (n-r) expression for number of digits from 2 to maxDigits var allTerms = List.filled(maxDigits-1, null) for (r in 2..maxDigits) {
var terms = []
pow = pow * 10
var pow1 = pow
var pow2 = 1
var i1 = 0
var i2 = r - 1
while (i1 < i2) {
terms.add(Term.new(pow1-pow2, i1, i2))
pow1 = (pow1/10).floor
pow2 = pow2 * 10
i1 = i1 + 1
i2 = i2 - 1
}
allTerms[r-2] = terms
}
// map of first minus last digits for 'n' to pairs giving this value var fml = {
0: [[2, 2], [8, 8]], 1: [[6, 5], [8, 7]], 4: 4, 0, 6: [[6, 0], [8, 2]]
}
// map of other digit differences for 'n' to pairs giving this value var dmd = {} for (i in 0...100) {
var a = [(i/10).floor, i%10]
var d = a[0] - a[1]
if (dmd[d]) {
dmd[d].add(a)
} else {
dmd[d] = [a]
}
} var fl = [0, 1, 4, 6] var dl = seq.call(-9, 9, 1) // all differences var zl = [0] // zero differences only var el = seq.call(-8, 8, 2) // even differences only var ol = seq.call(-9, 9, 2) // odd differences only var il = seq.call(0, 9, 1) var rares = [] var lists = List.filled(4, null) for (i in 0..3) lists[i] = [[fl[i]]] var digits = [] var count = 0
// Recursive closure to generate (n+r) candidates from (n-r) candidates // and hence find Rare numbers with a given number of digits. var fnpr fnpr = Fn.new { |cand, di, dis, indices, nmr, nd, level|
if (level == dis.count) {
digits[indices[0][0]] = fml[cand[0]][di[0]][0]
digits[indices[0][1]] = fml[cand[0]][di[0]][1]
var le = di.count
if (nd%2 == 1) {
le = le - 1
digits[(nd/2).floor] = di[le]
}
var i = 0
for (d in di[1...le]) {
digits[indices[i+1][0]] = dmd[cand[i+1]][d][0]
digits[indices[i+1][1]] = dmd[cand[i+1]][d][1]
i = i + 1
}
var r = toInt.call(digits, true)
var npr = nmr + 2*r
if (!isSquare.call(npr)) return
count = count + 1
Fmt.write(" R/N $2d:", count)
var ms = ((System.clock - start)*1000).round
Fmt.write(" $,7d ms", ms)
var n = toInt.call(digits, false)
Fmt.print(" ($,d)", n)
rares.add(n)
} else {
for (num in dis[level]) {
di[level] = num
fnpr.call(cand, di, dis, indices, nmr, nd, level+1)
}
}
}
// Recursive closure to generate (n-r) candidates with a given number of digits. var fnmr fnmr = Fn.new { |cand, list, indices, nd, level|
if (level == list.count) {
var nmr = 0
var nmr2 = 0
var i = 0
for (t in allTerms[nd-2]) {
if (cand[i] >= 0) {
nmr = nmr + t.coeff*cand[i]
} else {
nmr2 = nmr2 - t.coeff*cand[i]
if (nmr >= nmr2) {
nmr = nmr - nmr2
nmr2 = 0
} else {
nmr2 = nmr2 - nmr
nmr = 0
}
}
i = i + 1
}
if (nmr2 >= nmr) return
nmr = nmr - nmr2
if (!isSquare.call(nmr)) return
var dis = []
dis.add(seq.call(0, fml[cand[0]].count-1, 1))
for (i in 1...cand.count) {
dis.add(seq.call(0, dmd[cand[i]].count-1, 1))
}
if (nd%2 == 1) dis.add(il)
var di = List.filled(dis.count, 0)
fnpr.call(cand, di, dis, indices, nmr, nd, 0)
} else {
for (num in list[level]) {
cand[level] = num
fnmr.call(cand, list, indices, nd, level+1)
}
}
}
for (nd in 2..maxDigits) {
digits = List.filled(nd, 0)
if (nd == 4) {
lists[0].add(zl)
lists[1].add(ol)
lists[2].add(el)
lists[3].add(ol)
} else if(allTerms[nd-2].count > lists[0].count) {
for (i in 0..3) lists[i].add(dl)
}
var indices = []
for (t in allTerms[nd-2]) indices.add([t.ix1, t.ix2])
for (list in lists) {
var cand = List.filled(list.count, 0)
fnmr.call(cand, list, indices, nd, 0)
}
var ms = ((System.clock - start)*1000).round
Fmt.print(" $2s digits: $,7d ms", nd, ms)
}
Sort.quick(rares) Fmt.print("\nThe rare numbers with up to $d digits are:\n", maxDigits) var i = 0 for (rare in rares) {
Fmt.print(" $2d: $,21d", i+1, rare)
i = i + 1
}</lang>
- Output:
Aggregate timings to process all numbers up to:
R/N 1: 0 ms (65)
2 digits: 0 ms
3 digits: 0 ms
4 digits: 0 ms
5 digits: 0 ms
R/N 2: 3 ms (621,770)
6 digits: 3 ms
7 digits: 8 ms
8 digits: 54 ms
R/N 3: 56 ms (281,089,082)
9 digits: 77 ms
R/N 4: 80 ms (2,022,652,202)
R/N 5: 355 ms (2,042,832,002)
10 digits: 1,044 ms
11 digits: 1,742 ms
R/N 6: 4,892 ms (872,546,974,178)
R/N 7: 5,270 ms (872,568,754,178)
R/N 8: 11,318 ms (868,591,084,757)
12 digits: 15,797 ms
R/N 9: 20,673 ms (6,979,302,951,885)
13 digits: 28,328 ms
R/N 10: 79,833 ms (20,313,693,904,202)
R/N 11: 81,291 ms (20,313,839,704,202)
R/N 12: 101,982 ms (20,331,657,922,202)
R/N 13: 104,914 ms (20,331,875,722,202)
R/N 14: 113,507 ms (20,333,875,702,202)
R/N 15: 287,986 ms (40,313,893,704,200)
R/N 16: 290,036 ms (40,351,893,720,200)
14 digits: 340,736 ms
R/N 17: 342,140 ms (200,142,385,731,002)
R/N 18: 345,178 ms (221,462,345,754,122)
R/N 19: 384,744 ms (816,984,566,129,618)
R/N 20: 405,787 ms (245,518,996,076,442)
R/N 21: 409,179 ms (204,238,494,066,002)
R/N 22: 410,135 ms (248,359,494,187,442)
R/N 23: 414,341 ms (244,062,891,224,042)
R/N 24: 511,124 ms (403,058,392,434,500)
R/N 25: 514,313 ms (441,054,594,034,340)
15 digits: 565,126 ms
The rare numbers with up to 15 digits are:
1: 65
2: 621,770
3: 281,089,082
4: 2,022,652,202
5: 2,042,832,002
6: 868,591,084,757
7: 872,546,974,178
8: 872,568,754,178
9: 6,979,302,951,885
10: 20,313,693,904,202
11: 20,313,839,704,202
12: 20,331,657,922,202
13: 20,331,875,722,202
14: 20,333,875,702,202
15: 40,313,893,704,200
16: 40,351,893,720,200
17: 200,142,385,731,002
18: 204,238,494,066,002
19: 221,462,345,754,122
20: 244,062,891,224,042
21: 245,518,996,076,442
22: 248,359,494,187,442
23: 403,058,392,434,500
24: 441,054,594,034,340
25: 816,984,566,129,618
Turbo
Ruffles the feathers a little with a time 5 times quicker than the 'traditional' version. <lang ecmascript>import "/sort" for Sort import "/fmt" for Fmt import "/date" for Date
class Z2 {
construct new(value, hasValue) {
_value = value
_hasValue = hasValue
}
value { _value }
hasValue { _hasValue }
}
var Pow10 = List.filled(16, 0)
var init = Fn.new {
Pow10[0] = 1 for (i in 1..15) Pow10[i] = 10 * Pow10[i-1]
}
var acc = 0 var bs = List.filled(100000, false) var L var H
var izRev izRev = Fn.new { |n, i, g|
if ((i/Pow10[n-1]).floor != g%10) return false if (n < 2) return true return izRev.call(n-1, i%Pow10[n-1], (g/10).floor)
}
var fG = Fn.new { |n, start, end, reset, step|
var i = step * start
var g = step * end
var e = step * reset
return Fn.new {
while (i < g) {
acc = acc + step
i = i + step
return Z2.new(acc, true)
}
i = e
acc = acc - (g - e)
return n.call()
}
}
class ZP {
construct new(n, g) {
_n = n
_g = g
}
n { _n }
g { _g }
}
class NLH {
construct new(e) {
var even = []
var odd = []
var n = e.n
var g = e.g
var i = n.call()
while (i.hasValue) {
for (p in g) {
var ng = p[0]
var gg = p[1]
if (ng > 0 || i.value > 0) {
var w = ng*Pow10[4] + gg + i.value
var ws = w.sqrt.floor
if (ws*ws == w) {
if (w%2 == 0) {
even.add(w)
} else {
odd.add(w)
}
}
}
}
i = n.call()
}
_even = even
_odd = odd
}
even { _even }
odd { _odd }
}
var makeL = Fn.new { |n|
var g = List.filled((n/2).floor - 3, null)
g[0] = Fn.new { Z2.new(0, false) }
var i = 1
while (i < (n/2).floor - 3) {
var s = -9
if (i == (n/2).floor - 4) s = -10
var l = Pow10[n-i-4] - Pow10[i+3]
acc = acc + l*s
g[i] = fG.call(g[i-1], s, 9, -9, l)
i = i + 1
}
var g0 = 0
var g1 = 0
var g2 = 0
var g3 = 0
var l0 = Pow10[n-5]
var l1 = Pow10[n-6]
var l2 = Pow10[n-7]
var l3 = Pow10[n-8]
var f = Fn.new {
var w = []
while (g0 < 7) {
var nn = g3*l3 + g2*l2 + g1*l1 + g0*l0
var gg = -1000*g3 - 100*g2 - 10*g1 - g0
if (g3 < 9) {
g3 = g3 + 1
} else {
g3 = -9
if (g2 < 9) {
g2 = g2 + 1
} else {
g2 = -9
if (g1 < 9) {
g1 = g1 + 1
} else {
g1 = -9
if (g0 == 1) g0 = 3
g0 = g0 + 1
}
}
}
if (bs[(Pow10[10]+gg)%10000]) w.add([nn, gg])
}
return w
}
return ZP.new(g[(n/2).floor-4], f.call())
}
var makeH = Fn.new { |n|
acc = -(Pow10[(n/2).floor] + Pow10[((n-1)/2).floor])
var g = List.filled(((n+1)/2).floor - 3, null)
g[0] = Fn.new { Z2.new(0, false) }
var i = 1
while (i < (n/2).floor - 3) {
var j = 0
if (i == ((n+1)/2).floor - 3) j = -1
g[i] = fG.call(g[i-1], j, 18, 0, Pow10[n-i-4]+Pow10[i+3])
if (n%2 == 1) {
g[((n+1)/2).floor-4] = fG.call(g[(n/2).floor-4], -1, 9, 0, 2*Pow10[(n/2).floor])
}
i = i + 1
}
var g0 = 4
var g1 = 0
var g2 = 0
var g3 = 0
var l0 = Pow10[n-5]
var l1 = Pow10[n-6]
var l2 = Pow10[n-7]
var l3 = Pow10[n-8]
var f = Fn.new {
var w = []
while (g0 < 17) {
var nn = g3*l3 + g2*l2 + g1*l1 + g0*l0
var gg = 1000*g3 + 100*g2 + 10*g1 + g0
if (g3 < 18) {
g3 = g3 + 1
} else {
g3 = 0
if (g2 < 18) {
g2 = g2 + 1
} else {
g2 = 0
if (g1 < 18) {
g1 = g1 + 1
} else {
g1 = 0
if (g0 == 6 || g0 == 9) g0 = g0 + 3
g0 = g0 + 1
}
}
}
if (bs[gg%10000]) w.add([nn, gg])
}
return w
}
return ZP.new(g[((n+1)/2).floor-4], f.call())
}
var rare = Fn.new { |n|
acc = 0
for (g in 0...10000) bs[(g*g)%10000] = true
L = NLH.new(makeL.call(n))
H = NLH.new(makeH.call(n))
var rares = []
for (l in L.even) {
for (h in H.even) {
var r = ((h - l)/2).floor
var z = h - r
if (izRev.call(n, r, z)) rares.add(z)
}
}
for (l in L.odd) {
for (h in H.odd) {
var r = ((h - l)/2).floor
var z = h - r
if (izRev.call(n, r, z)) rares.add(z)
}
}
if (rares.count > 0) Sort.quick(rares)
return rares
}
// Formats time in form hh:mm:ss.fff (i.e. millisecond precision). var formatTime = Fn.new { |d|
var ms = (d * 1000).round var tm = Date.fromNumber(ms) Date.default = Date.isoTime + "|.|ttt" return tm.toString
}
var bStart = System.clock // block time var tStart = bStart // total time init.call() var nth = 3 // i.e. count of rare numbers < 10 digits System.print("nth rare number digs block time total time") for (nd in 10..15) {
var rares = rare.call(nd)
if (rares.count > 0) {
var i = 0
for (r in rares) {
nth = nth + 1
var t = ""
if (i < rares.count - 1) t = "\n"
Fmt.write("$2d $,21d$s", nth, r, t)
i = i + 1
}
} else {
Fmt.write("$26s", "")
}
var fbTime = formatTime.call(System.clock - bStart)
var ftTime = formatTime.call(System.clock - tStart)
Fmt.print(" $2d: $s $s", nd, fbTime, ftTime)
bStart = System.clock // restart block timing
}</lang>
- Output:
nth rare number digs block time total time
4 2,022,652,202
5 2,042,832,002 10: 00:00:00.084 00:00:00.084
11: 00:00:00.262 00:00:00.346
6 868,591,084,757
7 872,546,974,178
8 872,568,754,178 12: 00:00:00.714 00:00:01.060
9 6,979,302,951,885 13: 00:00:04.601 00:00:05.662
10 20,313,693,904,202
11 20,313,839,704,202
12 20,331,657,922,202
13 20,331,875,722,202
14 20,333,875,702,202
15 40,313,893,704,200
16 40,351,893,720,200 14: 00:00:13.764 00:00:19.426
17 200,142,385,731,002
18 204,238,494,066,002
19 221,462,345,754,122
20 244,062,891,224,042
21 245,518,996,076,442
22 248,359,494,187,442
23 403,058,392,434,500
24 441,054,594,034,340
25 816,984,566,129,618 15: 00:01:34.206 00:01:53.632