This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A216671 #30 Aug 11 2015 09:17:01
%S A216671 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 A216671 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 A216671 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 A216671 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 A216671 "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 A216671 This statement is only true for k = 1, 2, 3.
%C A216671 For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.
%C A216671 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 A216671 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 A216671 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 A216671 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 A216671 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 A216671 David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989. - From _N. J. A. Sloane_, Sep 14 2012
%F A216671 The fraction of terms with a(n)>0 goes to zero as n increases. - _Charles R Greathouse IV_, Sep 11 2012
%o A216671 (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 A216671 Cf. A000290, A001481, A002479, A003136, A020670, A216451, A216500.
%Y A216671 Cf. A154777, A092572.
%K A216671 nonn
%O A216671 1,1
%A A216671 _V. Raman_, Sep 13 2012
%E A216671 Edited by _N. J. A. Sloane_, Sep 14 2012