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

25, 842, 2306, 2402, 2459, 3602, 3650, 3803, 6081, 6242, 6338, 6779, 7058, 7319, 7643, 8088, 8354, 8363, 8402, 8543, 8761, 9122, 10607, 10826, 11257, 11378, 11447, 12203, 12458, 12722, 12984, 13273, 13682, 14162, 14424, 14639, 14738, 15362, 15626, 15698, 16475, 16634

Offset: 1

Author

Christopher E. Thompson, Jun 24 2016

Keywords

• nonn

Comments

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:

(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 odd.

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.

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.

Links

• Christopher E. Thompson, Table of n, a(n) for n = 1..727 [values up to 250000]
• H. T. Freitag, G. M. Phillips, On the sum of consecutive squares, Proc. 6th Intl Res. Conf. Fib. Numbers, Pullman, July 18-22, 1994 (1996), p 137.

Crossrefs

Cf. A001032, A134419, A274471.