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

564, 842, 1284, 2306, 2308, 2402, 2459, 3602, 3650, 3803, 6242, 6338, 6779, 7044, 7058, 7319, 7643, 8088, 8354, 8363, 8402, 8543, 8628, 9122, 9168, 9412, 10607, 10826, 10852, 11257, 11378, 11447, 12203, 12436, 12458, 12722, 12984, 13682, 14162, 14388, 14424, 14639

Offset: 1

Author

Christopher E. Thompson, Jun 24 2016

Keywords

• nonn

Comments

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:

(defining x||y to mean x|y and x and y/x are coprime)

if 3^e||n with e>0, then e is odd and (n/3^e)=2 (mod 3);

if p^e||n with p=5 or 7 (mod 12), then e is even;

if 3^e||(n+1) with e>0, then e is odd;

if p^e||(n+1) with p=3 (mod 4) and p>3, then e is even.

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.

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.

Links

• Christopher E. Thompson, Table of n, a(n) for n = 1..799 [values up to 250000]

Crossrefs

Cf. A134419, A274469, A274470.