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A274471 Numbers missing from A134419 despite satisfying the necessary congruence conditions (see comments).

%I A274471 #19 Feb 28 2018 08:43:07
%S A274471 564,842,1284,2306,2308,2402,2459,3602,3650,3803,6242,6338,6779,7044,
%T A274471 7058,7319,7643,8088,8354,8363,8402,8543,8628,9122,9168,9412,10607,
%U A274471 10826,10852,11257,11378,11447,12203,12436,12458,12722,12984,13682,14162,14388,14424,14639
%N A274471 Numbers missing from A134419 despite satisfying the necessary congruence conditions (see comments).
%C A274471 A134419 consists of those n where x^2 - n*y^2 = n(n-1)(n+1)/3 has integer solutions for x and y. There are easily verified necessary congruence conditions for that to occur:
%C A274471   (defining x||y to mean x|y and x and y/x are coprime)
%C A274471   if 3^e||n with e>0, then e is odd and (n/3^e)=2 (mod 3);
%C A274471   if p^e||n with p=5 or 7 (mod 12), then e is even;
%C A274471   if 3^e||(n+1) with e>0, then e is odd;
%C A274471   if p^e||(n+1) with p=3 (mod 4) and p>3, then e is even.
%C A274471 However, these conditions are not sufficient. This sequence consists of the numbers n satisfying the congruence conditions but for which the Pellian equation has no integer solutions.
%C A274471 If n = k^2*m where m is squarefree, then a necessary (but not sufficient) condition for n to occur in this sequence is that the narrow class group of quadratic forms of discriminant 4*m has more than one class per genus, or equivalently that the narrow class group is not an elementary 2-group.
%H A274471 Christopher E. Thompson, <a href="/A274471/b274471.txt">Table of n, a(n) for n = 1..799</a> [values up to 250000]
%Y A274471 Cf. A134419, A274469, A274470.
%K A274471 nonn
%O A274471 1,1
%A A274471 _Christopher E. Thompson_, Jun 24 2016