cp's OEIS Frontend

That is, numbers n such that Sum_{i=n..k} i^2 is a square for some k > n.

The paper by Bremner, Stroeker, and Tzanakis describes how they found all n <= 100 by solving elliptic curves. Their solutions are the same as the terms in this sequence. They also show that there are only a finite number of sums of squares beginning with n^2 that sum to a square. For example, starting with 3^2, there are only 3 ways to sum consecutive squares to produce a square: 3^2 + 4^2, 3^2 + ... + 580^2, and 3^2 + ... + 963^2. See A184762, A184763, A184885, and A184886 for more results from their paper.

This sequence is more difficult than A001032, which has the possible lengths of the sequences of consecutive squares that sum to a square. Be careful adding terms to this sequence; a simple search may miss some terms. An elliptic curve needs to be solved for each number.

It is conjectured that the sequence continues 103, 104, 106, 112, 115, 117, 119, 121, 124, 128, 129, 131, 132, 137, 140, 142, 146, 158, 168, 170, 172, 175, 178, 181, 183, 192, 193, 197, 199, 200. - Jean-François Alcover, Sep 17 2013. Conjecture confirmed (see the Schoenfield link below). - Jon E. Schoenfield, Nov 22 2013

For a(1) see A000330.

The corresponding squares are in A075405, the numbers of terms in the sum = a(n)-n+1 are in A075406.

All terms were verified by solving elliptic curves. If a(n)>0, then there may be additional values of m that produce squares. See A184763 for more information.

It is an old result (see Watson) that for n=1 the only k>n is k=24. Bremner, Stroeker, and Tzanakis compute the k for n <= 100 by solving elliptic curves. This sequence lists the number of k for each n; the values of k are in A184763. Sequence A180442 lists the n for which a(n) is nonzero.