A216680 - OEIS

A216680

Numbers 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.

%I #10 Sep 16 2012 08:58:24

%S 1,14,15,30,35,42,46,47,55,60,62,69,70,78,87,94,95,105,110,115,119,

%T 120,126,135,138,140,141,142,143,154,155,158,159,165,167,168,174,182,

%U 186,188,190,195,206,210,213,215,220,222,230,231,235,238,240,248,254,255,266,270,276,280,282,285,286,287,295,299

%N Numbers 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.

%C 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.

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

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

%C Essentially the same as A216679. - _R. J. Mathar_, Sep 16 2012

%Y Cf. A216451, A216500.

%K nonn

%O 1,2

%A _V. Raman_, Sep 13 2012