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%I #14 Oct 16 2012 11:28:50
%S 0,1,1,1,1,1,1,2,1,1,2,2,2,0,0,2,2,2,2,1,1,1,1,1,1,1,1,1,2,0,1,2,1,2,
%T 0,2,3,1,1,1,2,0,3,2,1,0,0,2,1,1,1,2,2,1,0,1,2,1,1,0,2,0,1,2,1,1,3,2,
%U 0,0,1,3,3,1,1,2,1,0,2,1,1,2,1,1,1,1,0,2,2,1,1,1,1,0,0,1,3,1,2,2,1,1,1,1,0,1,2,2,3,0,1,2,3,1,0,2,2,1,0,0
%N Let S_k = {x^2+k*y^2: x,y positive integers}. How many out of S_1, S_2, S_3, S_7 does n belong to?
%C "If a composite number C is of the form a^2 + kb^2 for some integers a & b, then every prime factor P (of C) raised to an odd power is of the form c^2 + kd^2, for some integers c & d."
%C 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.
%C A number can be written as a^2 + b^2 if and only if it has no prime factor congruent to 3 (mod 4) raised to an odd power.
%C A number can be written as a^2 + 2b^2 if and only if it has no prime factor congruent to 5 (mod 8) or 7 (mod 8) raised to an odd power.
%C A number can be written as a^2 + 3b^2 if and only if it has no prime factor congruent to 2 (mod 3) raised to an odd power.
%C A number can be written as a^2 + 7b^2 if and only if it has no prime factor congruent to 3 (mod 7) or 5 (mod 7) or 6 (mod 7) raised to an odd power and the exponent of 2 is not 1.
%F a(n) = 0 for almost all n. - _Charles R Greathouse IV_, Sep 14 2012
%o (PARI) for(n=1, 100, sol=0; for(x=1, 100, if(issquare(n-x*x)&&n-x*x>0, sol++; break)); for(x=1, 100, if(issquare(n-2*x*x)&&n-2*x*x>0, sol++; break)); for(x=1, 100, if(issquare(n-3*x*x)&&n-3*x*x>0, sol++; break)); for(x=1, 100, if(issquare(n-7*x*x)&&n-7*x*x>0, sol++; break)); print1(sol", ")) /* _V. Raman_, Oct 16 2012 */
%Y Cf. A000290, A001481, A002479, A003136, A020670, A216451, A216500, A154777, A092572.
%K nonn
%O 1,8
%A _V. Raman_, Sep 07 2012
%E Edited by _N. J. A. Sloane_, Sep 11 2012
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