A217869 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).
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, 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, 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
Offset: 1
Keywords
References
- David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
Crossrefs
Cf. A217463 (related sequence of this when the order does not matter for the equation a^2 + b^2 = n).
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).
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).
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).
Cf. A216278 (number of solutions to n = a^2+2*b^2 with a > 0, b > 0).
Cf. A092573 (number of solutions to n = a^2+3*b^2 with a > 0, b > 0).
Cf. A216511 (number of solutions to n = a^2+7*b^2 with a > 0, b > 0).
Programs
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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", "))
Comments