A216408 Perfect squares which can be written neither as a^2+b^2, nor as a^2+2*b^2, nor as a^2+3*b^2, nor as a^2+7*b^2, with a > 0 and b > 0.

1, 2209, 27889, 96721, 146689, 229441, 253009, 418609, 516961, 703921, 786769, 966289, 1324801, 1495729, 1739761, 2211169, 2283121, 2430481, 3323329, 3411409, 4255969, 4879681, 5527201, 5755201, 7091569, 7219969, 8427409, 8994001, 9138529, 10029889, 10182481, 11282881, 11607649, 12439729, 13476241, 14922769, 15295921

Offset: 1

Author

V. Raman, Sep 17 2012

Keywords

• nonn

Comments

If a composite number C, in case, can be written in the form C = a^2+k*b^2, for some integers a & b, then every prime factor P (for C) being raised to an odd power can be written in the form P = c^2+k*d^2, for some integers c & d.

This statement is only true for k = 1, 2, 3.

For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.

Crossrefs

Cf. A216451, A216500, A216501, A216671, A216679, A216680, A216682