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).

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, 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, 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

Offset: 1

Author

V. Raman, Oct 04 2012

Keywords

• nonn

Comments

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.

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.

1, 2, 3, 7 are the first four numbers, with the class number 1.

"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."

This statement is only true for k = 1, 2, 3.

For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.

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.

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.

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.

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.

References

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

Crossrefs

Cf. A216501, A216671.

Cf. A217869 (related sequence of this when the order does 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. 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).

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

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","))