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A274469 Numbers missing from A001032 despite satisfying the necessary congruence conditions (see comments).

%I A274469 #27 Feb 24 2018 12:07:57
%S A274469 25,842,2306,2402,2459,3602,3650,3803,6081,6242,6338,6779,7058,7319,
%T A274469 7643,8088,8354,8363,8402,8543,8761,9122,10607,10826,11257,11378,
%U A274469 11447,12203,12458,12722,12984,13273,13682,14162,14424,14639,14738,15362,15626,15698,16475,16634
%N A274469 Numbers missing from A001032 despite satisfying the necessary congruence conditions (see comments).
%C A274469 A001032 consists of those n for which there is a sequence of n consecutive positive squares whose sum is a square. For the associated Pellian equation, see A134419. The necessary congruence conditions described in A274471 apply here:
%C A274469   (defining x||y to mean x|y and x and y/x are coprime)
%C A274469   if 3^e||n with e>0, then e is odd and (n/3^e)=2 (mod 3);
%C A274469   if p^e||n with p=5 or 7 (mod 12), then e is even;
%C A274469   if 3^e||(n+1) with e>0, then e is odd;
%C A274469   if p^e||(n+1) with p=3 (mod 4) and p>3, then e is even.
%C A274469 In addition, in order that the Pellian equation has solutions of the correct parity, one must have:
%C A274469   if 2^e||n with e>0, then e is odd.
%C A274469 However, these conditions are not sufficient. This sequence consists of the numbers n that satisfy all the congruence conditions but for which there is no sequence of n consecutive positive squares whose sum is a square.
%C A274469 The term 25 is present despite the Pellian equation having a solution with the correct parity, because it leads only to 0^2 + 1^2 + ... + 24^2 = 70^2 and the specification of A001032 requires the squares to be strictly positive. (One may wonder whether this is quite natural, compare A185545.) In every other case the Pellian equation lacks solutions with the right parity. Note however that it may still have solutions with the opposite parity (this can happen only if n=1 mod 8) and so this sequence is not a subsequence of A274471.
%H A274469 Christopher E. Thompson, <a href="/A274469/b274469.txt">Table of n, a(n) for n = 1..727</a> [values up to 250000]
%H A274469 H. T. Freitag, G. M. Phillips, <a href="http://dx.doi.org/10.1007/978-94-009-0223-7_12">On the sum of consecutive squares</a>, Proc. 6th Intl Res. Conf. Fib. Numbers, Pullman, July 18-22, 1994 (1996), p 137.
%Y A274469 Cf. A001032, A134419, A274471.
%K A274469 nonn
%O A274469 1,1
%A A274469 _Christopher E. Thompson_, Jun 24 2016