A217869 - OEIS

%I #19 Feb 16 2024 06:31:48

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

%T 0,2,4,1,1,2,3,0,3,2,2,0,0,2,1,3,2,5,3,2,0,1,3,2,1,0,3,0,1,3,4,2,3,3,

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

%N 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 matters for the equation a^2+b^2 = n).

%C Note: For the equation a^2 + b^2 = n, if there are two solutions (a,b) and (b,a), then they will be counted separately.

%C 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 1, 2, 3, 7 are the first four numbers, with the class number 1. a(n) = A217463(n), when n is not the sum of two positive squares.

%C But when n is the sum of two positive squares, the ordered pairs for the equation x^2+y^2 = n count.

%C For example,

%C 193 = 12^2 + 7^2.

%C 193 = 7^2 + 12^2.

%C 193 = 11^2 + 2*6^2.

%C 193 = 1^2 + 3*8^2.

%C 193 = 9^2 + 7*4^2.

%C So, a(193) = 5. On the other hand, for the sequence A217463, the ordered pairs 12^2 + 7^2, 7^2 + 12^2 will be counted only once, so A217463(193) = 4.

%D David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

%o (PARI) for(n=1, 100, sol=0; for(x=1, 100, if(issquare(n-x*x)&&n-x*x>0, 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 Cf. A216501, A216671.

%Y Cf. A217463 (related sequence of this when the order does not matter for the equation a^2 + b^2 = n).

%Y 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 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 Cf. A063725 (number of solutions to n = a^2+b^2 (when the solutions (a, b) and (b, a) are being counted differently) with a > 0, b > 0).

%Y Cf. A216278 (number of solutions to n = a^2+2*b^2 with a > 0, b > 0).

%Y Cf. A092573 (number of solutions to n = a^2+3*b^2 with a > 0, b > 0).

%Y Cf. A216511 (number of solutions to n = a^2+7*b^2 with a > 0, b > 0).

%K nonn

%O 1,5

%A _V. Raman_, Oct 13 2012