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A234287
Number of distinct quadratic forms of discriminant -4n by which some prime can be represented.
2
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
1,3
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).
CROSSREFS
KEYWORD
nonn
AUTHOR
V. Raman, Dec 22 2013
STATUS
approved