A218979 Numbers n such that some sum of n consecutive positive cubes is a square.

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, 40, 41, 42, 43, 45, 47, 48, 49, 50, 51, 53, 54, 55, 57, 59, 60, 61, 63, 64, 65, 67, 69, 71, 72, 73, 75, 76, 77, 79, 81, 82, 83, 85, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99

Offset

1,2

Comments

The trivial solutions with x = 0 and x = 1 are not considered here.

Numbers n such that x^3 + (x+1)^3 + ... + (x+n-1)^3 = y^2 has nontrivial solutions over the integers.

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]

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

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

Links

Table of n, a(n) for n=1..68.

Michel Marcus, Examples of triples up to n=50

Vladimir Pletser, Triplets (n, x, y) with n,x less than 10^5

Vladimir Pletser, Number of terms, first term and square root of sums of consecutive cubed integers equal to integer squares, Research Gate, 2015.

Vladimir Pletser, 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, Research Gate, 2015. See Reference 19.

V. Pletser, General solutions of sums of consecutive cubed integers equal to squared integers, arXiv:1501.06098 [math.NT], 2015.

R. J. Stroeker, On the sum of consecutive cubes being a perfect square, Compositio Mathematica, 97 no. 1-2 (1995), pp. 295-307.

Example

See "Examples of triples" link.

Prog

(PARI)
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)))
n=1; while(n<100, if(a(n), print1(n, ", ")); n++) \\ Derek Orr, Aug 30 2014

Crossrefs

Cf. A116108, A116145, A126200, A126203, A163392, A163393, A253679, A253680, A253681, A253707, A253708, A253709.

Sequence in context: A024352 A288525 A134407 * A325417 A183868 A346232

Adjacent sequences: A218976 A218977 A218978 * A218980 A218981 A218982

Keyword

nonn

Author

Michel Marcus, Nov 08 2012

Extensions

Name changed, a(1) = 1 prepended and a(39)-a(68) from Derek Orr, Aug 30 2014

More terms for 50<n<10^5 and 1<x<10^5 from Vladimir Pletser, Jan 10 2015

Status

approved