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A225909 Numbers that are both a sum of two positive cubes and a difference of two consecutive cubes.

%I A225909 #18 Sep 07 2018 20:09:22
%S A225909 91,217,1027,4921,8587,14911,31519,39331,106597,117019,136747,185257,
%T A225909 195841,265519,281827,616987,636181,684019,712969,724717,736561,
%U A225909 955981,1200169,1352737,1405621,1771777,2481571,2756167,2937331,4251871,4996171,5262901
%N A225909 Numbers that are both a sum of two positive cubes and a difference of two consecutive cubes.
%C A225909 Solutions x to the equations x = a^3 + b^3 = (c+1)^3 - c^3 in positive integers. The values of c are A226902.
%C A225909 The intersection of A003325 and A003215.
%C A225909 Subsequence of A225908 = numbers that are both a sum and a difference of two positive cubes.
%C A225909 Shiraishi's solution to Gokai Ampon's equation u^3 + v^3 + w^3 = n^3 (see A023042 and A226903) shows that the sequence is infinite.
%D A225909 Shiraishi Chochu (aka Shiraishi Nagatada), Shamei Sampu (Sacred Mathematics), 1826.
%H A225909 Chai Wah Wu, <a href="/A225909/b225909.txt">Table of n, a(n) for n = 1..5000</a>
%H A225909 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 A225909 Wikipedia (French), <a href="http://fr.wikipedia.org/wiki/Shiraishi_Nagatada">Shiraishi Nagatada</a>
%H A225909 Wikipedia (German), <a href="http://de.wikipedia.org/wiki/Shiraishi_Nagatada">Shiraishi Nagatada</a>
%H A225909 <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%F A225909 a(n) = (A226902(n)+1)^3 - A226902(n)^3.
%e A225909 3^3 + 4^3 = 6^3 - 5^3 = 91, so 91 is a member.
%t A225909 Select[#[[2]]-#[[1]]&/@Partition[Range[5000]^3,2,1],Count[ IntegerPartitions[ #,{2}],_?(AllTrue[Surd[#,3],IntegerQ]&)]>0&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Sep 07 2018 *)
%Y A225909 Cf. A003215, A003325, A023042, A181123, A225908, A226902, A226903.
%K A225909 nonn
%O A225909 1,1
%A A225909 _Jonathan Sondow_, Jun 21 2013