A075404 Smallest m > n such that Sum_{i=n..m} i^2 is a square, or 0 if no such m exists.

24, 0, 4, 0, 0, 0, 29, 0, 32, 0, 22908, 0, 108, 0, 111, 0, 39, 28, 0, 21, 116, 80, 0, 0, 48, 0, 59, 77, 0, 198, 0, 609, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 67, 0, 0, 0, 0, 0, 171, 0, 147, 0, 0, 3533, 0, 0, 2132, 0, 92, 0, 0, 0, 305, 282, 0, 116, 0, 0, 0, 0, 0, 194, 36554, 0, 99, 0, 0, 0, 0, 0, 0, 276, 0, 0, 0, 136, 0, 0, 0, 332, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

1,1

Comments

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.

References

See A180442.

Links

Table of n, a(n) for n=1..100.

Example

a(1) = 24 because 1^2+...+24^2 = 70^2, a(7) = 29 because 7^2+...+29^2 = 92^2.

Mathematica

s[n_, k_]:=Module[{m=n+k-1}, (m(m+1)(2m+1)-n(n-1)(2n-1))/6]; mx=40000; Table[k=2; While[k<mx && !IntegerQ[Sqrt[s[n, k]]], k++]; If[k==mx, 0, n+k-1], {n, 100}]

Crossrefs

Cf. A000330, A075405, A075406, A180442 (n such that a(n) > 0).

Sequence in context: A357967 A353227 A075406 * A356304 A308234 A362795

Adjacent sequences: A075401 A075402 A075403 * A075405 A075406 A075407

Keyword

nonn

Author

Zak Seidov, Sep 13 2002

Extensions

Corrected and extended by Lior Manor Sep 19 2002

Corrected and edited by T. D. Noe, Jan 21 2011

Status

approved