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
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 10,000^th 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.

REXX

<lang rexx>/*REXX program displays the first N (lowest) taxicab numbers. */ parse arg N L1 H1 L2 H2 . /*obtain the optional numbers. */ if N== | N==',' then N=25 /*N given? Then use the default.*/ if L1== | L1==',' then L1=1 /*L1 " " " " " */ if H1== | H1==',' then H1=N /*H1 " " " " " */ if L2== | L2==',' then L2=10000 /*L2 " " " " " */ if H2== | H2==',' then H2=10007 /*H2 " " " " " */ mx=max(L1, H1, L2, H2) /*find how many taxicab #s needed*/ w=length(mx) /*width is used formatting output*/ numeric digits 20 /*prepare to use larger numbers. */

=0; @.=0; $.=; $=; b='■'; t='**3' /*initialize some REXX variables.*/

                                      /* [↓]  generate extra taxicab #s*/
do j=1   until  zz==mx                /*taxicab nums won't be in order.*/
jjj=j**3                              /*might as well calculate a cube.*/
z=;      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)t"■■■+"right(k,6,b)t
                         iterate      /* ··· and then go and do another*/
                         end          /* [↑] define one cube at a time.*/
         z=s                          /*save cube for  taxicab #  list.*/
         @.s=@.s","right(j,9,b)t"■■■+"right(k,6,b)t   /*build the text.*/
         $.s=right(s,15,b)'■■■'@.s    /*define the rest of taxicab #s. */
         leave                        /*leave the      DO  K    loop.  */
         end   /*k*/                  /* [↑]  build cubes one-at-a-time*/
$=$ z                                 /*define a   #.   taxicab number.*/
zz=words($)                           /*count the number of taxicab #s.*/
end   /*j*/                           /* [↑]  complete with overage #s.*/
                                      /*sort sub builds array, sorts it*/

list=esort(zz) /*sort taxicab nums, create list.*/

                                      /* [↓]  list  N  taxicab numbers.*/
    do j=L1  to H1;    _=word(list,j) /*get one taxicab num at a time. */
    say right(j,w)  translate($._,,b) /*display taxicab # (with blanks)*/
    end   /*j*/                       /* [↑] ■ are translated to blanks*/

say

    do j=L2  to H2;    _=word(list,j) /*get one taxicab num at a time. */
    say right(j,w)  translate($._,,b) /*display taxicab # (with blanks)*/
    end   /*j*/                       /* [↑] ■ are translated to blanks*/

exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────ESORT subroutine────────────────────*/ esort: procedure expose #. $; parse arg N 1 h a

      do j=1  for N;   #.j=word($,j);  end   /*j*/
      do  while  h>1;  h=h%2;          do i=1  for N-h; j=i; k=h+i
      do  while  #.k<#.j;     t=#.j;   #.j=#.k;  #.k=t
          if h>=j  then leave;  j=j-h;   k=k-h;  end;  end;  end;
      do m=1  for  N;  a=a #.m;  end  /*m*/;                     return a</lang>

output using the input: 1 25 0 0

 
 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           32832   ────►    30**3   +    18**3,       32**3   +     4**3
 5           39312   ────►    33**3   +    15**3,       34**3   +     2**3
 6           46683   ────►    30**3   +    27**3,       36**3   +     3**3
 7           65728   ────►    33**3   +    31**3,       40**3   +    12**3
 8          110808   ────►    45**3   +    27**3,       48**3   +     6**3
 9          134379   ────►    43**3   +    38**3,       51**3   +    12**3
10          149389   ────►    50**3   +    29**3,       53**3   +     8**3
11          171288   ────►    54**3   +    24**3,       55**3   +    17**3
12          195841   ────►    57**3   +    22**3,       58**3   +     9**3
13          216027   ────►    59**3   +    22**3,       60**3   +     3**3
14          314496   ────►    66**3   +    30**3,       68**3   +     4**3
15          402597   ────►    61**3   +    56**3,       69**3   +    42**3
16          443889   ────►    73**3   +    38**3,       76**3   +    17**3
17          513000   ────►    75**3   +    45**3,       80**3   +    10**3
18          513856   ────►    72**3   +    52**3,       78**3   +    34**3
19          558441   ────►    72**3   +    57**3,       81**3   +    30**3
20          593047   ────►    70**3   +    63**3,       84**3   +     7**3
21          684019   ────►    75**3   +    64**3,       82**3   +    51**3
22          704977   ────►    86**3   +    41**3,       89**3   +     2**3
23          805688   ────►    92**3   +    30**3,       93**3   +    11**3
24          886464   ────►    90**3   +    54**3,       96**3   +    12**3
25         1016496   ────►    90**3   +    66**3,       97**3   +    47**3