cp's OEIS Frontend

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

%I A075404 #23 Apr 16 2025 05:35:40
%S A075404 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,
%T A075404 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,
%U A075404 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
%N A075404 Smallest m > n such that Sum_{i=n..m} i^2 is a square, or 0 if no such m exists.
%C A075404 For a(1) see A000330.
%C A075404 The corresponding squares are in A075405, the numbers of terms in the sum = a(n)-n+1 are in A075406.
%C A075404 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.
%D A075404 See A180442.
%e A075404 a(1) = 24 because 1^2+...+24^2 = 70^2, a(7) = 29 because 7^2+...+29^2 = 92^2.
%t A075404 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}]
%Y A075404 Cf. A000330, A075405, A075406, A180442 (n such that a(n) > 0).
%K A075404 nonn
%O A075404 1,1
%A A075404 _Zak Seidov_, Sep 13 2002
%E A075404 Corrected and extended by _Lior Manor_, Sep 19 2002
%E A075404 Corrected and edited by _T. D. Noe_, Jan 21 2011