A234287 Number of distinct quadratic forms of discriminant -4n by which some prime can be represented.

1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 4, 4, 2, 3, 5, 3, 4, 4, 3, 4, 4, 3, 4, 4, 3, 5, 5, 2, 4, 4, 5, 5, 4, 3, 5, 5, 3, 4, 5, 4, 5, 5, 4, 4, 5, 4, 7, 4, 2, 6, 5, 4, 5, 5, 4, 6, 6, 3, 6, 6, 4, 5, 6, 3, 6, 6, 5, 6, 4, 4, 7, 5, 3, 6, 7, 4, 6, 5, 5, 7, 7, 5, 5, 4, 5, 6, 7, 3, 6, 6, 5

Offset: 1

Author

V. Raman, Dec 22 2013

Keywords

• nonn

Comments

This is similar to A232551, except that this includes non-primitive quadratic forms like 2x^2+2xy+4y^2 and 2x^2+4y^2 because the prime 2 can be represented by both of them. But unlike A067752, we do not include quadratic forms like 4x^2+2xy+4y^2 and 4x^2+4xy+4y^2 by which no prime can be represented.

So, when n == 3 (mod 4), this includes the additional non-primitive quadratic form 2x^2+2xy+((n+1)/2)y^2 and when p^2 divides n, where p is prime, this includes the additional non-primitive quadratic form px^2+(n/p)y^2.

If p is a prime and if p^2 does not divide n, then there exist a unique non-primitive quadratic form of discriminant = -4n by which p can be represented if and only if -n is a quadratic residue (mod p) and there exists a multiple of p which can be written in the form x^2+ny^2 in which p appears raised to an odd power, except when p = 2 and n == 3 (mod 8).

Links

• V. Raman, Examples of these distinct quadratic forms for n = 1..100

Crossrefs

Cf. A000003, A000926, A067752, A232550, A232551.