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 2,000^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.

J

<lang J> 25 {. ;({."#. <@(0&#`{.@.(1<#))/. ])/:~~.,/(+,/:~@,)"0/~3^~1+i.100

 1729     1   1728
 4104     8   4096
13832     8  13824
20683  1000  19683
32832    64  32768
39312     8  39304
40033   729  39304
46683    27  46656
64232  4913  59319
65728  1728  64000

110656 64 110592 110808 216 110592 134379 1728 132651 149389 512 148877 165464 8000 157464 171288 4913 166375 195841 729 195112 216027 27 216000 216125 125 216000 262656 512 262144 314496 64 314432 320264 5832 314432 327763 27000 300763 373464 216 373248 402597 74088 328509</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.

Python

(Magic number 1201 found by trial and error) <lang python></lang>

Output:

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=4000 /*L2 " " " " " */ if H2== | H2==',' then H2=4007 /*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. */

1. =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. */
         $=$ z                        /*define a   #.   taxicab number.*/
         zz=words($)                  /*count the number of taxicab #s.*/
         end   /*k*/                  /* [↑]  build cubes one-at-a-time*/
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 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      1802609991          1002**3   +   927**3,     1056**3   +   855**3
2001      1836207171          1134**3   +   723**3,     1155**3   +   666**3
2002      1876948136          1050**3   +   896**3,     1148**3   +   714**3
2003      1898380125          1090**3   +   845**3,     1125**3   +   780**3
2004      1948631048          1010**3   +   972**3,     1154**3   +   744**3
2005      2027722248          1074**3   +   924**3,     1153**3   +   791**3
2006      2080128456          1058**3   +   964**3,     1136**3   +   850**3
2007      2134551608          1052**3   +   990**3,     1124**3   +   894**3