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

4, 564, 1284, 2308, 3601, 7044, 7057, 7513, 8628, 9168, 9412, 10561, 10852, 11257, 12436, 13897, 14113, 14388, 14425, 16144, 16692, 16753, 17124, 17257, 17737, 18064, 18433, 18708, 19408, 19428, 20628, 20688, 20752, 20788, 20977, 21073, 23668, 25153, 27193, 28212, 28228

Offset: 1

Author

Christopher E. Thompson, Jun 24 2016

Keywords

• nonn

Comments

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:

(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.

In addition, in order that the Pellian equation has solutions of the correct parity, one must have:

if 2^e||n with e>0, then e is even;

if n is odd, then n=1 (mod 8).

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.

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.

Links

• Christopher E. Thompson, Table of n, a(n) for n = 1..325 [values up to 250000]

Crossrefs

Cf. A001033, A134419, A274471.