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%I #26 May 15 2017 12:07:03
%S 91,152,189,217,513,728,1027,1216,1512,1736,2457,3087,4104,4706,4921,
%T 4977,5103,5256,5824,5859,6832,7657,8216,8587,9728,10712,11375,12096,
%U 12691,13851,13888,14911,15093,15561,16120,16263,19000,19656,21014,23058,23625,24696
%N Numbers that are both a sum and a difference of two positive cubes.
%C Solutions x to the equations x = a^3 + b^3 = c^3 - d^3 in positive integers.
%C The intersection of A003325 and A181123. See those sequences for additional comments, references, links and cross-refs.
%C Suggested by Shiraishi's solutions to Gokai Ampon's equation u^3 + v^3 + w^3 = n^3 (transpose a term from the left side to the right side). See A023042 and A226903.
%C An infinite subsequence is (A226904(n)+1)^3 - A226904(n)^3.
%D Shiraishi Chochu (aka Shiraishi Nagatada), Shamei Sampu (Sacred Mathematics), 1826.
%H Chai Wah Wu, <a href="/A225908/b225908.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H David Eugene Smith and Yoshio Mikami, <a href="http://archive.org/details/historyofjapanes00smitiala">A History of Japanese Mathematics</a>, Open Court, Chicago, 1914; Dover reprint, 2004; pp. 233-235.
%H Wikipedia (French), <a href="http://fr.wikipedia.org/wiki/Shiraishi_Nagatada">Shiraishi Nagatada</a>
%H Wikipedia (German), <a href="http://de.wikipedia.org/wiki/Shiraishi_Nagatada">Shiraishi Nagatada</a>
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%e 3^3 + 4^3 + 5^3 = 6^3, so 3^3 + 4^3 = 91 and 3^3 + 5^3 = 152 and 4^3 + 5^3 = 189 are members.
%t nn = 3*10^4; t1 = Union[Flatten[Table[x^3 + y^3, {x, nn^(1/3)}, {y, x, (nn - x^3)^(1/3)}]]]; p = 3; t2 = Union[Reap[Do[n = i^p - j^p; If[n <= nn, Sow[n]], {i, Ceiling[(nn/p)^(1/(p - 1))]}, {j, i}]][[2, 1]]]; Intersection[t1, t2] (* _T. D. Noe_, Jun 21 2013 *)
%Y Cf. A003325, A023042, A181123, A226903, A226904.
%K nonn
%O 1,1
%A _Jonathan Sondow_, Jun 21 2013
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