Taxicab numbers: Difference between revisions
(→{{header|Python}}: Add Python.) |
m (added another name for taxicab numbers.) |
||
Line 13: | Line 13: | ||
:::* taxicab numbers |
:::* taxicab numbers |
||
:::* taxi cab numbers |
:::* taxi cab numbers |
||
:::* taxi-cab numbers |
|||
:::* Hardy-Ramanujan numbers |
:::* Hardy-Ramanujan numbers |
||
Revision as of 10:10, 14 March 2014
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.
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>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])) #sum2cubes[c1 + c2].add(tuple(sorted([cube2n[c1], cube2n[c2]])))
taxied = sorted((k, v) for k,v in sum2cubes.items() if len(v) == 2)
- pp(len(taxied)) # 2068
pp(taxied[:25]) pp(taxied[2000-1:2000-1+6])</lang>
- Output:
[(1729, {(12, 1), (10, 9)}), (4104, {(16, 2), (15, 9)}), (13832, {(20, 18), (24, 2)}), (20683, {(27, 10), (24, 19)}), (32832, {(30, 18), (32, 4)}), (39312, {(33, 15), (34, 2)}), (40033, {(33, 16), (34, 9)}), (46683, {(30, 27), (36, 3)}), (64232, {(36, 26), (39, 17)}), (65728, {(33, 31), (40, 12)}), (110656, {(48, 4), (40, 36)}), (110808, {(48, 6), (45, 27)}), (134379, {(51, 12), (43, 38)}), (149389, {(50, 29), (53, 8)}), (165464, {(54, 20), (48, 38)}), (171288, {(54, 24), (55, 17)}), (195841, {(57, 22), (58, 9)}), (216027, {(60, 3), (59, 22)}), (216125, {(60, 5), (50, 45)}), (262656, {(64, 8), (60, 36)}), (314496, {(66, 30), (68, 4)}), (320264, {(66, 32), (68, 18)}), (327763, {(58, 51), (67, 30)}), (373464, {(72, 6), (60, 54)}), (402597, {(69, 42), (61, 56)})] [(1682523899, {(1187, 216), (1178, 363)}), (1685502000, {(1155, 525), (1190, 70)}), (1686234375, {(1020, 855), (1175, 400)}), (1686316625, {(1145, 570), (1190, 105)}), (1691183069, {(1164, 485), (1061, 792)}), (1691367643, {(1099, 714), (1155, 532)})]
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. */
- =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