cp's OEIS Frontend

A217463 a(n) is the sum of total number of positive integer solutions to each of a^2 + b^2 = n, a^2 + 2*b^2 = n, a^2 + 3*b^2 = n and a^2 + 7*b^2 = n. (Order does not matter for the equation a^2+b^2 = n).

%I A217463 #26 Feb 16 2024 06:32:05
%S A217463 0,1,1,1,1,1,1,2,1,1,2,2,2,0,0,2,2,2,2,1,1,1,1,1,1,1,2,3,2,0,1,3,2,2,
%T A217463 0,2,3,1,1,1,2,0,3,2,1,0,0,2,1,2,2,4,2,2,0,1,3,1,1,0,2,0,1,3,2,2,3,2,
%U A217463 0,0,1,3,3,1,1,4,1,0,2,1,2,2,1,3,2,1,0,3,2,1,2,1,1,0,0,1,3,1,4,2
%N A217463 a(n) is the sum of total number of positive integer solutions to each of a^2 + b^2 = n, a^2 + 2*b^2 = n, a^2 + 3*b^2 = n and a^2 + 7*b^2 = n. (Order does not matter for the equation a^2+b^2 = n).
%C A217463 Note: For the equation a^2 + b^2 = n, if there are two solutions (a,b) and (b,a), then they will be counted only once.
%C A217463 The sequences A216501 and A216671 give how many of the four k values, k = 1, 2, 3, 7 does the equation a^2 + k*b^2 = n have a solution to.
%C A217463 1, 2, 3, 7 are the first four numbers, with the class number 1.
%C A217463 "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 A217463 This statement is only true for k = 1, 2, 3.
%C A217463 For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.
%C A217463 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 A217463 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 A217463 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 A217463 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.
%D A217463 David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
%o A217463 (PARI) for(n=1,100,sol=0;for(x=1,100,if(issquare(n-x*x)&&n-x*x>0&&x*x<=n-x*x,sol++);if(issquare(n-2*x*x)&&n-2*x*x>0,sol++);if(issquare(n-3*x*x)&&n-3*x*x>0,sol++);if(issquare(n-7*x*x)&&n-7*x*x>0,sol++));printf(sol","))
%Y A217463 Cf. A216501, A216671.
%Y A217463 Cf. A217869 (related sequence of this when the order does matter for the equation a^2 + b^2 = n).
%Y A217463 Cf. A216501 (how many of the four k values, k = 1, 2, 3, 7 does the equation a^2 + k*b^2 = n have a solution to, with a > 0, b > 0).
%Y A217463 Cf. A216671 (how many of the four k values, k = 1, 2, 3, 7 does the equation a^2 + k*b^2 = n have a solution to, with a >= 0, b >= 0).
%Y A217463 Cf. A025426 (number of solutions to n = a^2+b^2 (when the solutions (a, b) and (b, a) are being counted as the same) with a > 0, b > 0).
%Y A217463 Cf. A216278 (number of solutions to n = a^2+2*b^2 with a > 0, b > 0).
%Y A217463 Cf. A092573 (number of solutions to n = a^2+3*b^2 with a > 0, b > 0).
%Y A217463 Cf. A216511 (number of solutions to n = a^2+7*b^2 with a > 0, b > 0).
%K A217463 nonn
%O A217463 1,8
%A A217463 _V. Raman_, Oct 04 2012