cp's OEIS Frontend

A216682 Perfect squares which can be written in all the four forms a^2+b^2, a^2+2*b^2, a^2+3*b^2 and a^2+7*b^2, with a > 0 and b > 0.

%I A216682 #6 Sep 15 2012 00:51:07
%S A216682 3600,4624,12100,12321,14400,18496,20449,24336,26896,30276,32400,
%T A216682 37249,41616,46225,48400,49284,51076,57600,73984,75076,81796,85264,
%U A216682 90000,97344,101124,106929,107584,108900,110889,112225,113569,115600,121104,126736,129600,139876,144400,148225,148996,150544,165649,166464,176400,184041,184900,193600,197136
%N A216682 Perfect squares which can be written in all the four forms a^2+b^2, a^2+2*b^2, a^2+3*b^2 and a^2+7*b^2, with a > 0 and b > 0.
%C A216682 If a composite number C, say, 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 A216682 This statement is only true for k = 1, 2, 3.
%C A216682 For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.
%Y A216682 Cf. A216451, A216500.
%K A216682 nonn
%O A216682 1,1
%A A216682 _V. Raman_, Sep 13 2012