A216827 Numbers whose squares 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, 47, 167, 311, 383, 479, 503, 647, 719, 839, 887, 983, 1151, 1223, 1319, 1487, 1511, 1559, 1823, 1847, 2063, 2209, 2351, 2399, 2663, 2687, 2903, 2999, 3023, 3167, 3191, 3359, 3407, 3527, 3671, 3863, 3911, 4007, 4079, 4583, 4679, 4703, 4751, 4871, 4919, 5039, 5087, 5351, 5519, 5591, 5711, 5879, 5927

Offset: 1

Author

V. Raman, Sep 17 2012

Keywords

• nonn

Comments

If a composite number C can be written in the form C = a^2+k*b^2, for some integers a and 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 and 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.