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A234287 Number of distinct quadratic forms of discriminant -4n by which some prime can be represented. 2

%I

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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.

%C 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).

%H V. Raman, <a href="/A234287/a234287.txt">Examples of these distinct quadratic forms for n = 1..100</a>

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

%K nonn

%O 1,3

%A _V. Raman_, Dec 22 2013

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Last modified January 22 07:39 EST 2020. Contains 331139 sequences. (Running on oeis4.)