I'm working on modernizing Rosetta Code's infrastructure. Starting with communications. Please accept this time-limited open invite to RC's Slack.. --Michael Mol (talk) 20:59, 30 May 2020 (UTC)

Rare numbers

From Rosetta Code
Task
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   (nr)
  •   (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



C#[edit]

Traditional[edit]

Translation of: Go

Converted to unsigned longs in order to reach 19 digits.

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(); }
}
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[edit]

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.

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
}
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++[edit]

The task[edit]

The following is a simple implementation that demonstrates the principle of Talk:Rare_numbers#A_few_more_mins.. Interestingly it compile with both g++ and clang++, but g++ produces incorrect output. It is sufficient to meet the unambitious requirements of this task.

 
// Rare Numbers : Nigel Galloway - December 20th., 2019
#include <iostream>
#include <functional>
#include <cmath>
#include <numeric>
constexpr std::array<const long,18> pow10{1,10,100,1000,10000,100000,1000000,10000000,100000000,1000000000,10000000000,100000000000,1000000000000,10000000000000,100000000000000,1000000000000000,10000000000000000,100000000000000000};
constexpr auto r1=([]{std::array<bool,10>n{}; for(auto g:{0,1,4,5,6,9}) n[g]=true; return n;})();
constexpr bool izRev( int n, long i, long g){return (i/pow10[n-1]!=g%10)? false : (n<2)? true : izRev(n-1,i%pow10[n-1],g/10);}
struct nLH{
std::vector<long>even{};
std::vector<long>odd{};
nLH(std::function<std::optional<long>()> g){while (auto n=g()){if (([n]{long g=sqrt(*n); return (g*g==n);})()) (*n%2==0)? even.push_back(*n) : odd.push_back(*n);}}
};
template<int z>void Rare(){
auto L=nLH(([n=std::array<int,z/2>{},p=([]{int g{z/2}; std::array<long,z/2>n{}; for(auto& n:n){n=pow10[z-g]-pow10[g-1]; --g;} return n;})()]() mutable{
for (auto g=n.begin();g<(n.end());++g) if (*g<9){*g+=1; while(!r1[(n[z/2-1]*9)%10]) *g+=1; return std::optional{std::inner_product(n.begin(),n.end(),p.begin(),0L)};} else *g=-9;
return std::optional<long>{};}));
auto H=nLH(([n=([]{std::array<int,(z+1)/2>n{}; *(n.end()-1)=1; *n.begin()-=1; return n;})(),
p=([]{int g{z/2}; std::array<long,(z+1)/2>n{}; for(auto& i:n) if (z%2==1&n[0]==0) i=2*pow10[z/2]; else {i=pow10[z-g]+pow10[g-1]; --g;} return n;})()]() mutable{
for (auto g=n.begin();g<(n.end());++g) if (*g<19&(z%2==0|n[0]<10)){
*g+=1; while(!r1[n[(z+1)/2-1]%10]) *g+=1; return std::optional{std::inner_product(n.begin(),n.end(),p.begin(),0L)};} else *g=0;
return std::optional<long>{};}));
std::cout<<"Rare numbers of length "<<z<<std::endl;
for(auto l:L.even) for(auto h:H.even){long r=(h-l)/2; if(izRev(z,r,h-r)) std::cout<<"n="<<h-r<<" r="<<r<<" n-r="<<l<<" n+r="<<h<<std::endl;}
for(auto l:L.odd) for(auto h:H.odd) {long r=(h-l)/2; if(izRev(z,r,h-r)) std::cout<<"n="<<h-r<<" r="<<r<<" n-r="<<l<<" n+r="<<h<<std::endl;}
}
int main(){
Rare<2>();
Rare<3>();
Rare<4>();
Rare<5>();
Rare<6>();
Rare<7>();
Rare<8>();
Rare<9>();
Rare<10>();
Rare<11>();
Rare<12>();
Rare<13>();
Rare<14>();
Rare<15>();
Rare<16>();
}
 
Output:
Rare numbers of length 2
n=65 r=56 n-r=9 n+r=121
Rare numbers of length 3
Rare numbers of length 4
Rare numbers of length 5
Rare numbers of length 6
n=621770 r=77126 n-r=544644 n+r=698896
Rare numbers of length 7
Rare numbers of length 8
Rare numbers of length 9
n=281089082 r=280980182 n-r=108900 n+r=562069264
Rare numbers of length 10
n=2022652202 r=2022562202 n-r=90000 n+r=4045214404
n=2042832002 r=2002382402 n-r=40449600 n+r=4045214404
Rare numbers of length 11
Rare numbers of length 12
n=872546974178 r=871479645278 n-r=1067328900 n+r=1744026619456
n=872568754178 r=871457865278 n-r=1110888900 n+r=1744026619456
n=868591084757 r=757480195868 n-r=111110888889 n+r=1626071280625
Rare numbers of length 13
n=6979302951885 r=5881592039796 n-r=1097710912089 n+r=12860894991681
Rare numbers of length 14
n=20313693904202 r=20240939631302 n-r=72754272900 n+r=40554633535504
n=20313839704202 r=20240793831302 n-r=73045872900 n+r=40554633535504
n=20331657922202 r=20222975613302 n-r=108682308900 n+r=40554633535504
n=20331875722202 r=20222757813302 n-r=109117908900 n+r=40554633535504
n=20333875702202 r=20220757833302 n-r=113117868900 n+r=40554633535504
n=40313893704200 r=240739831304 n-r=40073153872896 n+r=40554633535504
n=40351893720200 r=202739815304 n-r=40149153904896 n+r=40554633535504
Rare numbers of length 15
n=200142385731002 r=200137583241002 n-r=4802490000 n+r=400279968972004
n=221462345754122 r=221457543264122 n-r=4802490000 n+r=442919889018244
n=816984566129618 r=816921665489618 n-r=62900640000 n+r=1633906231619236
n=245518996076442 r=244670699815542 n-r=848296260900 n+r=490189695891984
n=204238494066002 r=200660494832402 n-r=3577999233600 n+r=404898988898404
n=248359494187442 r=244781494953842 n-r=3577999233600 n+r=493140989141284
n=244062891224042 r=240422198260442 n-r=3640692963600 n+r=484485089484484
n=403058392434500 r=5434293850304 n-r=397624098584196 n+r=408492686284804
n=441054594034340 r=43430495450144 n-r=397624098584196 n+r=484485089484484
Rare numbers of length 16
n=2133786945766212 r=2126675496873312 n-r=7111448892900 n+r=4260462442639524
n=2135568943984212 r=2124893498655312 n-r=10675445328900 n+r=4260462442639524
n=8191154686620818 r=8180266864511918 n-r=10887822108900 n+r=16371421551132736
n=8191156864620818 r=8180264686511918 n-r=10892178108900 n+r=16371421551132736
n=2135764587964212 r=2124697854675312 n-r=11066733288900 n+r=4260462442639524
n=2135786765764212 r=2124675676875312 n-r=11111088888900 n+r=4260462442639524
n=8191376864400818 r=8180044686731918 n-r=11332177668900 n+r=16371421551132736
n=2078311262161202 r=2021612621138702 n-r=56698641022500 n+r=4099923883299904
n=4135786945764210 r=124675496875314 n-r=4011111448888896 n+r=4260462442639524
n=6889765708183410 r=143818075679886 n-r=6745947632503524 n+r=7033583783863296
n=8052956026592517 r=7152956206592508 n-r=899999820000009 n+r=15205912233185025
n=8052956206592517 r=7152956026592508 n-r=900000180000009 n+r=15205912233185025
n=8650327689541457 r=7541459867230568 n-r=1108867822310889 n+r=16191787556772025
n=8650349867341457 r=7541437689430568 n-r=1108912177910889 n+r=16191787556772025
n=6157577986646405 r=5046466897757516 n-r=1111111088888889 n+r=11204044884403921

Rare numbers of length 17
n=86965750494756968 r=86965749405756968 n-r=1089000000 n+r=173931499900513936
n=22542040692914522 r=22541929604024522 n-r=111088890000 n+r=45083970296939044
n=67725910561765640 r=4656716501952776 n-r=63069194059812864 n+r=72382627063718416

10 to 19 digits[edit]

The following is a faster implementation of the algorithm. It compiles with both g++ and clang++, both produce the correct output. It will not work for lengths less than 10 or greater than 19. The timings are using Mingw running on a Core I5 1035G1.

 
// Rare Numbers : Nigel Galloway - December 20th., 2019
#include <iostream>
#include <functional>
#include <bitset>
#include <cmath>
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};
constexpr bool izRev(int n,unsigned long i,unsigned 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 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<unsigned long>even{};
std::vector<unsigned long>odd{};
nLH(std::pair<Z1,std::vector<std::pair<long,long>>> e){auto [n,g]=e; while (auto i=n()){for(auto [ng,gg]:g){ if((ng>0)|(*i>0)){
unsigned long w=ng*pow10[4]+gg+*i; unsigned long g=sqrt(w); if(g*g==w) (w%2==0)? even.push_back(w) : odd.push_back(w);}}}}
};
class Rare{
long acc{0};
const std::bitset<10000>bs;
const std::pair<Z1,std::vector<std::pair<long,long>>> makeL(const int n){
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]);
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;})()}{
std::cout<<"Rare "<<n<<std::endl;
for(auto l:L.even) for(auto h:H.even){unsigned long r{(h-l)/2},z{(h-r)}; if(izRev(n,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) {unsigned long r{(h-l)/2},z{(h-r)}; if(izRev(n,r,z)) std::cout<<"n="<<z<<" r="<<r<<" n-r="<<l<<" n+r="<<h<<std::endl;}
}
};
int main(){
Rare(19);
}
 
Output:
Rare 10
n=2022652202 r=2022562202 n-r=90000 n+r=4045214404
n=2042832002 r=2002382402 n-r=40449600 n+r=4045214404
Rare 11
Rare 12
n=872546974178 r=871479645278 n-r=1067328900 n+r=1744026619456
n=872568754178 r=871457865278 n-r=1110888900 n+r=1744026619456
n=868591084757 r=757480195868 n-r=111110888889 n+r=1626071280625
Rare 13
n=6979302951885 r=5881592039796 n-r=1097710912089 n+r=12860894991681
Rare 14
n=20331657922202 r=20222975613302 n-r=108682308900 n+r=40554633535504
n=20331875722202 r=20222757813302 n-r=109117908900 n+r=40554633535504
n=40351893720200 r=202739815304 n-r=40149153904896 n+r=40554633535504
n=20313693904202 r=20240939631302 n-r=72754272900 n+r=40554633535504
n=20313839704202 r=20240793831302 n-r=73045872900 n+r=40554633535504
n=20333875702202 r=20220757833302 n-r=113117868900 n+r=40554633535504
n=40313893704200 r=240739831304 n-r=40073153872896 n+r=40554633535504
Rare 15
n=245518996076442 r=244670699815542 n-r=848296260900 n+r=490189695891984
n=204238494066002 r=200660494832402 n-r=3577999233600 n+r=404898988898404
n=248359494187442 r=244781494953842 n-r=3577999233600 n+r=493140989141284
n=200142385731002 r=200137583241002 n-r=4802490000 n+r=400279968972004
n=221462345754122 r=221457543264122 n-r=4802490000 n+r=442919889018244
n=441054594034340 r=43430495450144 n-r=397624098584196 n+r=484485089484484
n=403058392434500 r=5434293850304 n-r=397624098584196 n+r=408492686284804
n=244062891224042 r=240422198260442 n-r=3640692963600 n+r=484485089484484
n=816984566129618 r=816921665489618 n-r=62900640000 n+r=1633906231619236
Rare 16
n=2135568943984212 r=2124893498655312 n-r=10675445328900 n+r=4260462442639524
n=2078311262161202 r=2021612621138702 n-r=56698641022500 n+r=4099923883299904
n=8191154686620818 r=8180266864511918 n-r=10887822108900 n+r=16371421551132736
n=8191156864620818 r=8180264686511918 n-r=10892178108900 n+r=16371421551132736
n=6889765708183410 r=143818075679886 n-r=6745947632503524 n+r=7033583783863296
n=2135764587964212 r=2124697854675312 n-r=11066733288900 n+r=4260462442639524
n=2135786765764212 r=2124675676875312 n-r=11111088888900 n+r=4260462442639524
n=2133786945766212 r=2126675496873312 n-r=7111448892900 n+r=4260462442639524
n=4135786945764210 r=124675496875314 n-r=4011111448888896 n+r=4260462442639524
n=8191376864400818 r=8180044686731918 n-r=11332177668900 n+r=16371421551132736
n=8650327689541457 r=7541459867230568 n-r=1108867822310889 n+r=16191787556772025
n=8650349867341457 r=7541437689430568 n-r=1108912177910889 n+r=16191787556772025
n=8052956026592517 r=7152956206592508 n-r=899999820000009 n+r=15205912233185025
n=8052956206592517 r=7152956026592508 n-r=900000180000009 n+r=15205912233185025
n=6157577986646405 r=5046466897757516 n-r=1111111088888889 n+r=11204044884403921

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

real    0m33.328s
user    0m32.078s
sys     0m0.015s

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

real    1m32.945s
user    1m31.656s
sys     0m0.000s

Rare 19
n=6988066446726832640 r=462386276446608896 n-r=6525680170280223744 n+r=7450452723173441536
n=2060303819041450202 r=2020541409183030602 n-r=39762409858419600 n+r=4080845228224480804
n=2702373360882732072 r=2702372880633732072 n-r=480249000000 n+r=5404746241516464144
n=2551755006254571552 r=2551754526005571552 n-r=480249000000 n+r=5103509532260143104
n=8066308349502036608 r=8066302059438036608 n-r=6290064000000 n+r=16132610408940073216
n=2825378427312735282 r=2825372137248735282 n-r=6290064000000 n+r=5650750564561470564
n=8320411466598809138 r=8319088956641140238 n-r=1322509957668900 n+r=16639500423239949376
n=2042401829204402402 r=2042044029281042402 n-r=357799923360000 n+r=4084445858485444804
n=2420424089100600242 r=2420060019804240242 n-r=364069296360000 n+r=4840484108904840484
n=8197906905009010818 r=8180109005096097918 n-r=17797899912912900 n+r=16378015910105108736
n=8200756128308135597 r=7955318038216570028 n-r=245438090091565569 n+r=16156074166524705625
n=6531727101458000045 r=5400008541017271356 n-r=1131718560440728689 n+r=11931735642475271401

real    12m21.298s
user    12m19.890s
sys     0m0.030s

20+ digits[edit]

 
// 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);
}
 
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[edit]

Translation of: Go

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.

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));
}
}
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#[edit]

The Function[edit]

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

 
// 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)
 

43 down[edit]

 
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
 
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

Go[edit]

Traditional[edit]

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.

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: {{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] = [][]int8{{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))
}
}
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[edit]

Translation of: C#

Twice as quick as the first Go version though, despite being a faithful translation, a little slower than the C# and VB.NET versions.

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)
}
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[edit]

Translation of: C++

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.

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
}
}
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

Java[edit]

Translation of: Kotlin
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)));
}
}
}
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[edit]

Translation of: Go
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()
 
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[edit]

Translation of: D
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)))
}
}
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[edit]

not optimized[edit]

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.

Works with: langur version 0.8.11
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)

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).

val .reverse = f toNumber join reverse split .n

Phix[edit]

naive[edit]

Ridiculously slow, 90s just for the first 3.

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
Output:
{1,65,"0s"}
{2,621770,"0.2s"}
{3,281089082,"1 minute and 29s"}

advanced[edit]

Translation of: VB.NET
Translation of: Go

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.

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})
{{{9,0,1}},
{{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] = {{4, 0}}
-- fml[6] = {{8, 3}} -- (um? - needs longer lists, & that append(lists[4],dl) below)
fml[7] = {{6, 0}, {8, 2}}
-- sequence lists = {{{0}},{{1}},{{4}},{{5}},{{6}}}
sequence lists = {{{0}},{{1}},{{4}},{{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})
{{{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}},
{{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] = {{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()
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

REXX[edit]

(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).

/*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 $
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

Ruby[edit]

Translation of: Kotlin

Not sure where, but there seems to be a bug that introduces false rare numbers as output beyond what is shown

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()
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

Visual Basic .NET[edit]

Traditional[edit]

Translation of: C#
via
Translation of: Go
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.
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
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[edit]

Translation of: C#
(translation of the quicker version)

Performance is better, only about 4% slower than C#.

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

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[edit]

Translation of: Go
Library: Wren-sort
Library: Wren-fmt
Library: Wren-date

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[edit]

About 9.5 minutes to find the first 25 rare numbers.

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
}
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[edit]

Ruffles the feathers a little with a time 5 times quicker than the 'traditional' version.

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
}
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