A216671 - OEIS

%I #30 Aug 11 2015 09:17:01

%S 4,2,2,4,1,1,2,3,4,1,2,2,2,0,0,4,2,2,2,1,1,1,1,1,4,1,2,2,2,0,1,3,1,2,

%T 0,4,3,1,1,1,2,0,3,2,1,0,0,2,4,2,1,2,2,1,0,1,2,1,1,0,2,0,2,4,1,1,3,2,

%U 0,0,1,3,3,1,2,2,1,0,2,1,4,2,1,1,1,1,0,2,2,1,1,1,1,0,0,1,3,2,2,4,1,1,1,1,0,1,2,2,3,0,1,2,3,1,0,2,2,1,0,0

%N Let S_k = {x^2+k*y^2: x,y nonnegative 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 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.

%C 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.

%C Comment from _N. J. A. Sloane_, Sep 14 2012: S_1, S_2, S_3, S_7 are the first four quadratic forms with class number 1. (See Cox, for example.)

%D David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989. - From _N. J. A. Sloane_, Sep 14 2012

%F The fraction of terms with a(n)>0 goes to zero as n increases. - _Charles R Greathouse IV_, Sep 11 2012

%o (PARI) for(n=1, 100, sol=0; for(x=0, 100, if(issquare(n-x*x)&&n-x*x>=0, sol++; break)); for(x=0, 100, if(issquare(n-2*x*x)&&n-2*x*x>=0, sol++; break)); for(x=0, 100, if(issquare(n-3*x*x)&&n-3*x*x>=0, sol++; break)); for(x=0, 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.

%Y Cf. A154777, A092572.

%K nonn

%O 1,1

%A _V. Raman_, Sep 13 2012

%E Edited by _N. J. A. Sloane_, Sep 14 2012