

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
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OFFSET

1,3


COMMENTS

This is similar to A232551, except that this includes nonprimitive 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 nonprimitive quadratic form 2x^2+2xy+((n+1)/2)y^2 and when p^2 divides n, where p is prime, this includes the additional nonprimitive 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 nonprimitive 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

Table of n, a(n) for n=1..100.
V. Raman, Examples of these distinct quadratic forms for n = 1..100


CROSSREFS

Cf. A000003, A000926, A067752, A232550, A232551.
Sequence in context: A251141 A319696 A320111 * A084294 A067752 A229942
Adjacent sequences: A234284 A234285 A234286 * A234288 A234289 A234290


KEYWORD

nonn


AUTHOR

V. Raman, Dec 22 2013


STATUS

approved



