cp's OEIS Frontend

A274470 Numbers missing from A001033 despite satisfying the necessary congruence conditions (see comments).

%I A274470 #21 May 24 2023 03:56:43
%S A274470 4,564,1284,2308,3601,7044,7057,7513,8628,9168,9412,10561,10852,11257,
%T A274470 12436,13897,14113,14388,14425,16144,16692,16753,17124,17257,17737,
%U A274470 18064,18433,18708,19408,19428,20628,20688,20752,20788,20977,21073,23668,25153,27193,28212,28228
%N A274470 Numbers missing from A001033 despite satisfying the necessary congruence conditions (see comments).
%C A274470 A001033 consists of those n for which there is a sequence of n consecutive positive odd squares whose sum is a square. For the associated Pellian equation, see A134419. The necessary congruence conditions described in A274471 apply here:
%C A274470   (defining x||y to mean x|y and x and y/x are coprime)
%C A274470   if 3^e||n with e>0, then e is odd and (n/3^e)=2 (mod 3);
%C A274470   if p^e||n with p=5 or 7 (mod 12), then e is even;
%C A274470   if 3^e||(n+1) with e>0, then e is odd;
%C A274470   if p^e||(n+1) with p=3 (mod 4) and p>3, then e is even.
%C A274470 In addition, in order that the Pellian equation has solutions  of the correct parity, one must have:
%C A274470   if 2^e||n with e>0, then e is even;
%C A274470   if n is odd, then n=1 (mod 8).
%C A274470 However, these conditions are not sufficient. This sequence consists of the numbers n that satisfy all of the congruence conditions but for which there is no sequence of n consecutive positive odd squares whose sum is a square.
%C A274470 The term 4 is present despite the Pellian equation having a solution with the correct parity, because it leads only to (-1)^2 + 1^2 + 3^2 + 5^2 = 6^2, and the specification of A001033 disallows squares of negative numbers. 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 A274470 Christopher E. Thompson, <a href="/A274470/b274470.txt">Table of n, a(n) for n = 1..325</a> [values up to 250000]
%Y A274470 Cf. A001033, A134419, A274471.
%K A274470 nonn
%O A274470 1,1
%A A274470 _Christopher E. Thompson_, Jun 24 2016