Taxicab numbers
definition
A taxicab number (the definition that is being used here) is a positive integer that can be expressed as the sum of two positive cubes in more than one way.
The first taxicab number is 1729, which is:
- 13 + 123 and
- 93 + 103.
Taxicab numbers are also known as:
- taxi numbers
- taxicab numbers
- taxi cab numbers
- taxi-cab numbers
- Hardy-Ramanujan numbers
task requirements
The task requirements are:
- compute the lowest 25 taxicab numbers.
- exactly 25 taxicab numbers are to be shown (in numeric order).
- show the taxicab numbers as well as its constituent cubes.
- show all numbers in a very readable (aligned) format.
- show the 2,000th taxicab number + a half dozen more (extra credit)).
See also
- Sequence A001235 taxicab numbers on The On-Line Encyclopedia of Integer Sequences.
- Entry Hardy-Ramanujan Number on The Eric Weisstein's World of Mathematics (TM).
- Entry taxicab number on The Eric Weisstein's World of Mathematics (TM).
- Wiki entry Hardy-Ramanujan number.
C
Using a priority queue to emit sum of two cubs in order. It's reasonably fast and doesn't use excessive amount of memory (the heap is only at 245 length upon the 2006th taxi). <lang c>#include <stdio.h>
- include <stdlib.h>
typedef unsigned long long xint; typedef unsigned uint; typedef struct { uint x, y; // x > y always xint value; } sum_t;
xint *cube; uint n_cubes;
sum_t *pq; uint pq_len, pq_cap;
void add_cube(void) { uint x = n_cubes++; cube = realloc(cube, sizeof(xint) * (n_cubes + 1)); cube[n_cubes] = (xint) n_cubes*n_cubes*n_cubes; if (x < 2) return; // x = 0 or 1 is useless
if (++pq_len >= pq_cap) { if (!(pq_cap *= 2)) pq_cap = 2; pq = realloc(pq, sizeof(*pq) * pq_cap); }
sum_t tmp = (sum_t) { x, 1, cube[x] + 1 }; // upheap uint i, j; for (i = pq_len; i >= 1 && pq[j = i>>1].value > tmp.value; i = j) pq[i] = pq[j];
pq[i] = tmp; }
void next_sum(void) { redo: while (!pq_len || pq[1].value >= cube[n_cubes]) add_cube();
sum_t tmp = pq[0] = pq[1]; // pq[0] always stores last seen value if (++tmp.y >= tmp.x) { // done with this x; throw it away tmp = pq[pq_len--]; if (!pq_len) goto redo; // refill empty heap } else tmp.value += cube[tmp.y] - cube[tmp.y-1];
uint i, j; // downheap for (i = 1; (j = i<<1) <= pq_len; pq[i] = pq[j], i = j) { if (j < pq_len && pq[j+1].value < pq[j].value) ++j; if (pq[j].value >= tmp.value) break; } pq[i] = tmp; }
uint next_taxi(sum_t *hist) { do next_sum(); while (pq[0].value != pq[1].value);
uint len = 1; hist[0] = pq[0]; do { hist[len++] = pq[1]; next_sum(); } while (pq[0].value == pq[1].value);
return len; }
int main(void) { uint i, l; sum_t x[10]; for (i = 1; i <= 2006; i++) { l = next_taxi(x); if (25 < i && i < 2000) continue; printf("%4u:%10llu", i, x[0].value); while (l--) printf(" = %4u^3 + %4u^3", x[l].x, x[l].y); putchar('\n'); } return 0; }</lang>
- Output:
1: 1729 = 12^3 + 1^3 = 10^3 + 9^3 2: 4104 = 15^3 + 9^3 = 16^3 + 2^3 3: 13832 = 20^3 + 18^3 = 24^3 + 2^3 4: 20683 = 27^3 + 10^3 = 24^3 + 19^3 5: 32832 = 30^3 + 18^3 = 32^3 + 4^3 6: 39312 = 33^3 + 15^3 = 34^3 + 2^3 7: 40033 = 33^3 + 16^3 = 34^3 + 9^3 8: 46683 = 30^3 + 27^3 = 36^3 + 3^3 9: 64232 = 36^3 + 26^3 = 39^3 + 17^3 10: 65728 = 33^3 + 31^3 = 40^3 + 12^3 11: 110656 = 40^3 + 36^3 = 48^3 + 4^3 12: 110808 = 45^3 + 27^3 = 48^3 + 6^3 13: 134379 = 43^3 + 38^3 = 51^3 + 12^3 14: 149389 = 50^3 + 29^3 = 53^3 + 8^3 15: 165464 = 48^3 + 38^3 = 54^3 + 20^3 16: 171288 = 54^3 + 24^3 = 55^3 + 17^3 17: 195841 = 57^3 + 22^3 = 58^3 + 9^3 18: 216027 = 59^3 + 22^3 = 60^3 + 3^3 19: 216125 = 50^3 + 45^3 = 60^3 + 5^3 20: 262656 = 60^3 + 36^3 = 64^3 + 8^3 21: 314496 = 66^3 + 30^3 = 68^3 + 4^3 22: 320264 = 66^3 + 32^3 = 68^3 + 18^3 23: 327763 = 58^3 + 51^3 = 67^3 + 30^3 24: 373464 = 60^3 + 54^3 = 72^3 + 6^3 25: 402597 = 61^3 + 56^3 = 69^3 + 42^3 2000:1671816384 = 1168^3 + 428^3 = 944^3 + 940^3 2001:1672470592 = 1124^3 + 632^3 = 1187^3 + 29^3 2002:1673170856 = 1034^3 + 828^3 = 1164^3 + 458^3 2003:1675045225 = 1153^3 + 522^3 = 1081^3 + 744^3 2004:1675958167 = 1096^3 + 711^3 = 1159^3 + 492^3 2005:1676926719 = 1188^3 + 63^3 = 1095^3 + 714^3 2006:1677646971 = 990^3 + 891^3 = 1188^3 + 99^3
D
High Level Version
<lang d>void main() {
import std.stdio, std.range, std.algorithm,std.typecons,std.string;
const cubes = iota(1, 1201).map!(x => tuple(x, ulong(x)^^3)).array;
bool[Tuple!(const int, const int)][ulong] sum2cubes;
foreach (const p1; cubes)
foreach (const p2; cubes[p1[0] .. $])
sum2cubes[p1[1] + p2[1]][tuple(p2[0], p1[0])] = true;
const taxied = zip(sum2cubes.byKey, sum2cubes.byValue)
.filter!(kv => kv[1].length > 1).array
.schwartzSort!(kv => kv[0]).release;
foreach (const r; [[0, 25], [2000-1, 2000+6]]) {
foreach (immutable i, const t; taxied[r[0] .. r[1]])
writefln("%4d: %10d = %-(%s = %)", i + r[0] + 1, t[0],
t[1].byKey.map!q{"%4d^3 + %4d^3".format(a[])});
writeln;
}
}</lang>
- Output:
1: 1729 = 12^3 + 1^3 = 10^3 + 9^3 2: 4104 = 16^3 + 2^3 = 15^3 + 9^3 3: 13832 = 24^3 + 2^3 = 20^3 + 18^3 4: 20683 = 27^3 + 10^3 = 24^3 + 19^3 5: 32832 = 30^3 + 18^3 = 32^3 + 4^3 6: 39312 = 34^3 + 2^3 = 33^3 + 15^3 7: 40033 = 34^3 + 9^3 = 33^3 + 16^3 8: 46683 = 30^3 + 27^3 = 36^3 + 3^3 9: 64232 = 36^3 + 26^3 = 39^3 + 17^3 10: 65728 = 40^3 + 12^3 = 33^3 + 31^3 11: 110656 = 48^3 + 4^3 = 40^3 + 36^3 12: 110808 = 48^3 + 6^3 = 45^3 + 27^3 13: 134379 = 43^3 + 38^3 = 51^3 + 12^3 14: 149389 = 53^3 + 8^3 = 50^3 + 29^3 15: 165464 = 54^3 + 20^3 = 48^3 + 38^3 16: 171288 = 54^3 + 24^3 = 55^3 + 17^3 17: 195841 = 58^3 + 9^3 = 57^3 + 22^3 18: 216027 = 59^3 + 22^3 = 60^3 + 3^3 19: 216125 = 50^3 + 45^3 = 60^3 + 5^3 20: 262656 = 60^3 + 36^3 = 64^3 + 8^3 21: 314496 = 68^3 + 4^3 = 66^3 + 30^3 22: 320264 = 68^3 + 18^3 = 66^3 + 32^3 23: 327763 = 67^3 + 30^3 = 58^3 + 51^3 24: 373464 = 72^3 + 6^3 = 60^3 + 54^3 25: 402597 = 69^3 + 42^3 = 61^3 + 56^3 2000: 1671816384 = 944^3 + 940^3 = 1168^3 + 428^3 2001: 1672470592 = 1187^3 + 29^3 = 1124^3 + 632^3 2002: 1673170856 = 1164^3 + 458^3 = 1034^3 + 828^3 2003: 1675045225 = 1153^3 + 522^3 = 1081^3 + 744^3 2004: 1675958167 = 1159^3 + 492^3 = 1096^3 + 711^3 2005: 1676926719 = 1188^3 + 63^3 = 1095^3 + 714^3 2006: 1677646971 = 1188^3 + 99^3 = 990^3 + 891^3
Faster Version
<lang d>struct Taxicabs {
alias Xint = uint; // Or ulong or BigInt.
static struct Sum {
uint x, y;
Xint value;
}
Xint[] cube; Sum[] pq; uint pq_len;
void addCube() pure nothrow {
cube.length = cube.length ? cube.length + 1 : 2;
cube[$ - 1] = cast(Xint)(cube.length - 1) ^^ 3;
if (cube.length < 2)
return; // 0 or 1 is useless.
if (++pq_len >= pq.length)
pq.length = (pq.length == 0) ? 2 : (pq.length * 2);
auto tmp = Sum(cube.length - 2, 1, cube[cube.length - 2] + 1);
// Upheap.
uint i, j;
for (i = pq_len; i >= 1 && pq[j = i >> 1].value > tmp.value; i = j)
pq[i] = pq[j];
pq[i] = tmp; }
void nextSum() pure nothrow {
redo:
while (!pq_len || pq[1].value >= cube[$ - 1])
addCube();
Sum tmp = pq[0] = pq[1]; // pq[0] always stores last seen value.
if (++tmp.y >= tmp.x) { // Done with this x; throw it away.
tmp = pq[pq_len--];
if (!pq_len)
goto redo; // Refill empty heap.
} else
tmp.value += cube[tmp.y] - cube[tmp.y - 1];
uint i, j;
// Downheap.
for (i = 1; (j = i << 1) <= pq_len; pq[i] = pq[j], i = j) {
if (j < pq_len && pq[j + 1].value < pq[j].value)
++j;
if (pq[j].value >= tmp.value)
break;
}
pq[i] = tmp;
}
uint nextTaxi(size_t N)(ref Sum[N] hist) pure nothrow {
do {
nextSum();
} while (pq[0].value != pq[1].value);
uint len = 1;
hist[0] = pq[0];
do {
hist[len++] = pq[1];
nextSum();
} while (pq[0].value == pq[1].value);
return len; }
}
void main() nothrow {
import core.stdc.stdio;
Taxicabs t; Taxicabs.Sum[3] x;
foreach (immutable uint i; 1 .. 2007) {
uint l = t.nextTaxi(x);
if (i > 25 && i < 2000)
continue;
printf("%4u: %10lu", i, x[0].value);
while (l--)
printf(" = %4u^3 + %4u^3", x[l].x, x[l].y);
putchar('\n');
}
}</lang>
- Output:
1: 1729 = 12^3 + 1^3 = 10^3 + 9^3 2: 4104 = 15^3 + 9^3 = 16^3 + 2^3 3: 13832 = 20^3 + 18^3 = 24^3 + 2^3 4: 20683 = 27^3 + 10^3 = 24^3 + 19^3 5: 32832 = 30^3 + 18^3 = 32^3 + 4^3 6: 39312 = 33^3 + 15^3 = 34^3 + 2^3 7: 40033 = 33^3 + 16^3 = 34^3 + 9^3 8: 46683 = 30^3 + 27^3 = 36^3 + 3^3 9: 64232 = 36^3 + 26^3 = 39^3 + 17^3 10: 65728 = 33^3 + 31^3 = 40^3 + 12^3 11: 110656 = 40^3 + 36^3 = 48^3 + 4^3 12: 110808 = 45^3 + 27^3 = 48^3 + 6^3 13: 134379 = 43^3 + 38^3 = 51^3 + 12^3 14: 149389 = 50^3 + 29^3 = 53^3 + 8^3 15: 165464 = 48^3 + 38^3 = 54^3 + 20^3 16: 171288 = 54^3 + 24^3 = 55^3 + 17^3 17: 195841 = 57^3 + 22^3 = 58^3 + 9^3 18: 216027 = 59^3 + 22^3 = 60^3 + 3^3 19: 216125 = 50^3 + 45^3 = 60^3 + 5^3 20: 262656 = 60^3 + 36^3 = 64^3 + 8^3 21: 314496 = 66^3 + 30^3 = 68^3 + 4^3 22: 320264 = 66^3 + 32^3 = 68^3 + 18^3 23: 327763 = 58^3 + 51^3 = 67^3 + 30^3 24: 373464 = 60^3 + 54^3 = 72^3 + 6^3 25: 402597 = 61^3 + 56^3 = 69^3 + 42^3 2000: 1671816384 = 1168^3 + 428^3 = 944^3 + 940^3 2001: 1672470592 = 1124^3 + 632^3 = 1187^3 + 29^3 2002: 1673170856 = 1034^3 + 828^3 = 1164^3 + 458^3 2003: 1675045225 = 1153^3 + 522^3 = 1081^3 + 744^3 2004: 1675958167 = 1096^3 + 711^3 = 1159^3 + 492^3 2005: 1676926719 = 1188^3 + 63^3 = 1095^3 + 714^3 2006: 1677646971 = 990^3 + 891^3 = 1188^3 + 99^3
Run-time: about 0.07 seconds with ldc2 compiler.
J
<lang J> 25 {. ;({."#. <@(0&#`({.@{.(;,)<@}."1)@.(1<#))/. ])/:~~.,/(+,/:~@,)"0/~3^~1+i.100 ┌──────┬────────────┬─────────────┐ │1729 │1 1728 │729 1000 │ ├──────┼────────────┼─────────────┤ │4104 │8 4096 │729 3375 │ ├──────┼────────────┼─────────────┤ │13832 │8 13824 │5832 8000 │ ├──────┼────────────┼─────────────┤ │20683 │1000 19683 │6859 13824 │ ├──────┼────────────┼─────────────┤ │32832 │64 32768 │5832 27000 │ ├──────┼────────────┼─────────────┤ │39312 │8 39304 │3375 35937 │ ├──────┼────────────┼─────────────┤ │40033 │729 39304 │4096 35937 │ ├──────┼────────────┼─────────────┤ │46683 │27 46656 │19683 27000 │ ├──────┼────────────┼─────────────┤ │64232 │4913 59319 │17576 46656 │ ├──────┼────────────┼─────────────┤ │65728 │1728 64000 │29791 35937 │ ├──────┼────────────┼─────────────┤ │110656│64 110592 │46656 64000 │ ├──────┼────────────┼─────────────┤ │110808│216 110592 │19683 91125 │ ├──────┼────────────┼─────────────┤ │134379│1728 132651 │54872 79507 │ ├──────┼────────────┼─────────────┤ │149389│512 148877 │24389 125000 │ ├──────┼────────────┼─────────────┤ │165464│8000 157464 │54872 110592 │ ├──────┼────────────┼─────────────┤ │171288│4913 166375 │13824 157464 │ ├──────┼────────────┼─────────────┤ │195841│729 195112 │10648 185193 │ ├──────┼────────────┼─────────────┤ │216027│27 216000 │10648 205379 │ ├──────┼────────────┼─────────────┤ │216125│125 216000 │91125 125000 │ ├──────┼────────────┼─────────────┤ │262656│512 262144 │46656 216000 │ ├──────┼────────────┼─────────────┤ │314496│64 314432 │27000 287496 │ ├──────┼────────────┼─────────────┤ │320264│5832 314432 │32768 287496 │ ├──────┼────────────┼─────────────┤ │327763│27000 300763│132651 195112│ ├──────┼────────────┼─────────────┤ │373464│216 373248 │157464 216000│ ├──────┼────────────┼─────────────┤ │402597│74088 328509│175616 226981│ └──────┴────────────┴─────────────┘</lang>
Explanation:
First, generate 100 cubes.
Then, form a 3 column table of unique rows: sum, small cube, large cube
Then, gather rows where the first entry is the same. Keep the ones with at least two such entries.
Note that the cube root of the 25th entry is slightly smaller than 74, so testing against the first 100 cubes is more than sufficient.
Perl 6
This uses a pretty simple search algorithm that doesn't necessarily return the Taxicab numbers in order. Assuming we want all the Taxicab numbers within some range S to N, we'll search until we find N values. When we find the Nth value, we continue to search up to the cube root of the largest Taxicab number found up to that point. That ensures we will find all of them inside the desired range without needing to search arbitrarily or use magic numbers. Defaults to returning the Taxicab numbers from 1 to 25. Pass in a different start and end value if you want some other range. <lang perl6>sub MAIN ($start = 1, $end = 25) {
my %taxi; my $taxis = 0; my $terminate = 0;
for 1 .. * -> $c1 {
display( %taxi, $start, $end ) and exit if 0 < $terminate < $c1;
my $c = $c1 ** 3;
for 1 .. $c1 -> $c2 {
my $this = $c2 ** 3 + $c;
%taxi{$this}.push: [$c2, $c1];
$taxis++ if %taxi{$this}.elems == 2;
$terminate = %taxi.grep( { $_.value.elems > 1 } ).sort( +*.key )[*-1].key**(1/3)
if $taxis == $end and !$terminate;
}
}
}
sub display (%this_stuff, $start, $end) {
my $i = $start;
printf "%4d %10d =>\t%s\n", $i++, $_.key,
($_.value.map({ sprintf "%4d³ + %-s", $_[0], "$_[1]³" })).join: ",\t"
for %this_stuff.grep( { $_.value.elems > 1 } ).sort( +*.key )[$start-1..$end-1];
1;
}</lang>
- Output:
With no passed parameters (default)
1 1729 => 9³ + 10³, 1³ + 12³ 2 4104 => 9³ + 15³, 2³ + 16³ 3 13832 => 18³ + 20³, 2³ + 24³ 4 20683 => 19³ + 24³, 10³ + 27³ 5 32832 => 18³ + 30³, 4³ + 32³ 6 39312 => 15³ + 33³, 2³ + 34³ 7 40033 => 16³ + 33³, 9³ + 34³ 8 46683 => 27³ + 30³, 3³ + 36³ 9 64232 => 26³ + 36³, 17³ + 39³ 10 65728 => 31³ + 33³, 12³ + 40³ 11 110656 => 36³ + 40³, 4³ + 48³ 12 110808 => 27³ + 45³, 6³ + 48³ 13 134379 => 38³ + 43³, 12³ + 51³ 14 149389 => 29³ + 50³, 8³ + 53³ 15 165464 => 38³ + 48³, 20³ + 54³ 16 171288 => 24³ + 54³, 17³ + 55³ 17 195841 => 22³ + 57³, 9³ + 58³ 18 216027 => 22³ + 59³, 3³ + 60³ 19 216125 => 45³ + 50³, 5³ + 60³ 20 262656 => 36³ + 60³, 8³ + 64³ 21 314496 => 30³ + 66³, 4³ + 68³ 22 320264 => 32³ + 66³, 18³ + 68³ 23 327763 => 51³ + 58³, 30³ + 67³ 24 373464 => 54³ + 60³, 6³ + 72³ 25 402597 => 56³ + 61³, 42³ + 69³
With passed parameters 2000 2006:
2000 1671816384 => 940³ + 944³, 428³ + 1168³ 2001 1672470592 => 632³ + 1124³, 29³ + 1187³ 2002 1673170856 => 828³ + 1034³, 458³ + 1164³ 2003 1675045225 => 744³ + 1081³, 522³ + 1153³ 2004 1675958167 => 711³ + 1096³, 492³ + 1159³ 2005 1676926719 => 714³ + 1095³, 63³ + 1188³ 2006 1677646971 => 891³ + 990³, 99³ + 1188³
Python
(Magic number 1201 found by trial and error) <lang python>from collections import defaultdict from itertools import product from pprint import pprint as pp
cube2n = {x**3:x for x in range(1, 1201)} sum2cubes = defaultdict(set) for c1, c2 in product(cube2n, cube2n): if c1 >= c2: sum2cubes[c1 + c2].add((cube2n[c1], cube2n[c2]))
taxied = sorted((k, v) for k,v in sum2cubes.items() if len(v) >= 2)
- pp(len(taxied)) # 2068
for t in enumerate(taxied[:25], 1):
pp(t)
print('...') for t in enumerate(taxied[2000-1:2000+6], 2000):
pp(t)</lang>
- Output:
(1, (1729, {(12, 1), (10, 9)}))
(2, (4104, {(16, 2), (15, 9)}))
(3, (13832, {(20, 18), (24, 2)}))
(4, (20683, {(27, 10), (24, 19)}))
(5, (32832, {(30, 18), (32, 4)}))
(6, (39312, {(33, 15), (34, 2)}))
(7, (40033, {(33, 16), (34, 9)}))
(8, (46683, {(30, 27), (36, 3)}))
(9, (64232, {(36, 26), (39, 17)}))
(10, (65728, {(33, 31), (40, 12)}))
(11, (110656, {(48, 4), (40, 36)}))
(12, (110808, {(48, 6), (45, 27)}))
(13, (134379, {(51, 12), (43, 38)}))
(14, (149389, {(50, 29), (53, 8)}))
(15, (165464, {(54, 20), (48, 38)}))
(16, (171288, {(54, 24), (55, 17)}))
(17, (195841, {(57, 22), (58, 9)}))
(18, (216027, {(60, 3), (59, 22)}))
(19, (216125, {(60, 5), (50, 45)}))
(20, (262656, {(64, 8), (60, 36)}))
(21, (314496, {(66, 30), (68, 4)}))
(22, (320264, {(66, 32), (68, 18)}))
(23, (327763, {(58, 51), (67, 30)}))
(24, (373464, {(72, 6), (60, 54)}))
(25, (402597, {(69, 42), (61, 56)}))
...
(2000, (1671816384, {(1168, 428), (944, 940)}))
(2001, (1672470592, {(1187, 29), (1124, 632)}))
(2002, (1673170856, {(1164, 458), (1034, 828)}))
(2003, (1675045225, {(1153, 522), (1081, 744)}))
(2004, (1675958167, {(1159, 492), (1096, 711)}))
(2005, (1676926719, {(1188, 63), (1095, 714)}))
(2006, (1677646971, {(990, 891), (1188, 99)}))
REXX
<lang rexx>/*REXX program displays the first (lowest) taxicab numbers. */ parse arg L1 H1 L2 H2 . /*obtain the optional numbers. */ if L1== | L1==',' then L1=1 /*L1 " " " " " */ if H1== | H1==',' then H1=25 /*H1 " " " " " */ if L2== | L2==',' then L2=2000 /*L2 " " " " " */ if H2== | H2==',' then H2=2007 /*H2 " " " " " */ mx=max(L1, H1, L2, H2) /*find how many taxicab #s needed*/ mx=mx+mx%10 /*cushion, compensate for triples*/ w=length(mx) /*width is used formatting output*/ numeric digits 20 /*prepare to use larger numbers. */
- =0; @.=0; $.=; b='■'; p='**3' /*initialize some REXX variables.*/
/* [↓] generate extra taxicab #s*/
do j=1 until #>=mx /*taxicab nums won't be in order.*/
jjj=j**3 /*might as well calculate a cube.*/
do k=1 for j-1; s=jjj+k**3 /*define a whole bunch of cubes. */
if @.s==0 then do /*if cube not defined, then do it*/
@.s = "────►"right(j,6,b)p"■■■+"right(k,6,b)p
iterate /* ··· and then go and do another*/
end /* [↑] define one cube at a time.*/
comma=pos(',',@.s)\==0 /*has it has been defined before?*/
@.s=@.s","right(j,9,b)p"■■■+"right(k,6,b)p /*build the text.*/
$.s=right(s,15,b)'■■■'@.s /*define the rest of taxicab #s. */
if comma then iterate /*S is a triple (or better). */
#=#+1 /*bump the taxicab number count. */
#.#=s /*define a #. taxicab number.*/
end /*k*/ /* [↑] build cubes one-at-a-time*/
end /*j*/ /* [↑] complete with overage #s.*/
h=mx /*H= ½─way point for pivot sort. */
do while h>1; h=h%2; do i=1 for mx-h; j=i; k=h+i /*sort. */
do while #.k<#.j; _=#.j; #.j=#.k; #.k=_ /*sort. */
if h>=j then leave; j=j-h; k=k-h; end; end /*sort. */
end /*while h>1*/ /* [↑] sort the taxicab # array.*/
call show L1,H1 /*show 1st range of taxicab #s. */ call show L2,H2 /*show 2nd range of taxicab #s. */ exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────SHOW subroutine─────────────────────*/ show: parse arg low,high; if low==0 then return /*show bunch taxicab#s,*/
do t=low to high; _=#.t /*get single taxicab # at a time.*/
say right(t,w) translate($._,,b) /*display taxicab # (with blanks)*/
end /*t*/ /* [↑] ■ are translated to blanks*/
return</lang> output using the default input:
1 1729 10**3 + 9**3, 12**3 + 1**3 2 4104 15**3 + 9**3, 16**3 + 2**3 3 13832 20**3 + 18**3, 24**3 + 2**3 4 20683 24**3 + 19**3, 27**3 + 10**3 5 32832 30**3 + 18**3, 32**3 + 4**3 6 39312 33**3 + 15**3, 34**3 + 2**3 7 40033 33**3 + 16**3, 34**3 + 9**3 8 46683 30**3 + 27**3, 36**3 + 3**3 9 64232 36**3 + 26**3, 39**3 + 17**3 10 65728 33**3 + 31**3, 40**3 + 12**3 11 110656 40**3 + 36**3, 48**3 + 4**3 12 110808 45**3 + 27**3, 48**3 + 6**3 13 134379 43**3 + 38**3, 51**3 + 12**3 14 149389 50**3 + 29**3, 53**3 + 8**3 15 165464 48**3 + 38**3, 54**3 + 20**3 16 171288 54**3 + 24**3, 55**3 + 17**3 17 195841 57**3 + 22**3, 58**3 + 9**3 18 216027 59**3 + 22**3, 60**3 + 3**3 19 216125 50**3 + 45**3, 60**3 + 5**3 20 262656 60**3 + 36**3, 64**3 + 8**3 21 314496 66**3 + 30**3, 68**3 + 4**3 22 320264 66**3 + 32**3, 68**3 + 18**3 23 327763 58**3 + 51**3, 67**3 + 30**3 24 373464 60**3 + 54**3, 72**3 + 6**3 25 402597 61**3 + 56**3, 69**3 + 42**3 2000 1671816384 944**3 + 940**3, 1168**3 + 428**3 2001 1672470592 1124**3 + 632**3, 1187**3 + 29**3 2002 1673170856 1034**3 + 828**3, 1164**3 + 458**3 2003 1675045225 1081**3 + 744**3, 1153**3 + 522**3 2004 1675958167 1096**3 + 711**3, 1159**3 + 492**3 2005 1676926719 1095**3 + 714**3, 1188**3 + 63**3 2006 1677646971 990**3 + 891**3, 1188**3 + 99**3 2007 1680918365 1132**3 + 613**3, 1189**3 + 16**3