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%I #52 Jan 19 2019 04:14:58
%S 1,3,5,7,8,9,11,12,13,15,17,18,19,21,23,25,27,28,29,31,32,33,35,37,39,
%T 40,41,42,43,45,47,48,49,50,51,53,54,55,57,59,60,61,63,64,65,67,69,71,
%U 72,73,75,76,77,79,81,82,83,85,87,89,91,92,93,94,95,97,98,99
%N Numbers n such that some sum of n consecutive positive cubes is a square.
%C The trivial solutions with x = 0 and x = 1 are not considered here.
%C Numbers n such that x^3 + (x+1)^3 + ... + (x+n-1)^3 = y^2 has nontrivial solutions over the integers.
%C The nontrivial solutions are found by solving Y^2 = X^3 + d(n)*X with d(n) = n^2*(n^2-1)/4 (A006011), Y = n*y and X = n*x + (1/2)*n*(n-1). [Corrected by _Derek Orr_, Aug 30 2014]
%C x^3 + (x+1)^3 + ... + (x+n-1)^3 = y^2 can also be written as y^2 = n(x + (n-1)/2)(n(x + (n-1)/2) + x(x-1)). - _Vladimir Pletser_, Aug 30 2014
%C There are 892 triples (n,x,y), with n and x less than 10^5 (1 < n,x < 10^5), which are nontrivial solutions of x^3 + (x+1)^3 + ... + (x+n-1)^3 = y^2 (note that (n,x,y) corresponds to (M,a,c) in A253679, A253680, A253681, A253707, A253708, A253709, A253724, A253725). - _Vladimir Pletser_, Jan 10 2015
%H Michel Marcus, <a href="/A218979/a218979.txt">Examples of triples up to n=50</a>
%H Vladimir Pletser, <a href="/A218979/a218979_1.txt">Triplets (n, x, y) with n,x less than 10^5</a>
%H Vladimir Pletser, <a href="http://www.researchgate.net/profile/Vladimir_Pletser/publication/271272786">Number of terms, first term and square root of sums of consecutive cubed integers equal to integer squares</a>, Research Gate, 2015.
%H Vladimir Pletser, <a href="http://www.researchgate.net/profile/Vladimir_Pletser/publication/272820117">Fundamental solutions of the Pell equation X^2-(sigma^4-delta^4)Y^2=delta^4 for the first 45 solutions of the sums of consecutive cubed integers equalling integer squares</a>, Research Gate, 2015. See Reference 19.
%H V. Pletser, <a href="http://arxiv.org/abs/1501.06098">General solutions of sums of consecutive cubed integers equal to squared integers</a>, arXiv:1501.06098 [math.NT], 2015.
%H R. J. Stroeker, <a href="http://www.numdam.org/item?id=CM_1995__97_1-2_295_0">On the sum of consecutive cubes being a perfect square</a>, Compositio Mathematica, 97 no. 1-2 (1995), pp. 295-307.
%e See "Examples of triples" link.
%o (PARI)
%o a(n)=for(x=2,10^7, /* note this limit only generates the terms in the data section */ X = n*x + (1/2)*n*(n-1); d=n^2*(n^2-1)/4;if(issquare(X^3+d*X),return(x)))
%o n=1;while(n<100,if(a(n),print1(n,", "));n++) \\ _Derek Orr_, Aug 30 2014
%Y Cf. A116108, A116145, A126200, A126203, A163392, A163393, A253679, A253680, A253681, A253707, A253708, A253709.
%K nonn
%O 1,2
%A _Michel Marcus_, Nov 08 2012
%E Name changed, a(1) = 1 prepended and a(39)-a(68) from _Derek Orr_, Aug 30 2014
%E More terms for 50<n<10^5 and 1<x<10^5 from _Vladimir Pletser_, Jan 10 2015
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