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a(n) = one-fourth of the (negated) fundamental even discriminants D := b^2-4*a*c<0 of positive definite binary quadratic forms F=a*x^2+b*x*y+c*y^2. See A039957 for the odd numbers and A003657 for the combined even and odd numbers.

The asymptotic density of this sequence is 4/Pi^2 (A185199). - Amiram Eldar, Feb 23 2021

This is one of the bisections of sequence A343238. The other sequence is A344232.

This is a proper subsequence of A020669.

The primes in this sequence are given in A033205.

Discriminant Disc = -20 = -4*5. Class number h(-20) = A000003(5) = 2. The reduced primitive forms representing the two proper (determinant = +1) equivalence classes are the present principal form F1 = [1, 0, 5] and F2 = [2, 2, 3] treated in A344232.

A positive integer k is properly represented by some primitive form of Disc = -20 if and only if the congruence s^2 + 20 == 0 (mod 4*k) has a solution. See, e.g., Buell Proposition 41, p. 50, or Scholz-Schoeneberg Satz 74, p. 105. That is, x^2 + 5 == 0 (mod k), with s = 2*x. For the representative solutions x from {0, 1, ..., k-1}, with k from A343238, see A343239. These solutions x determine the so-called representative parallel primitive forms (rpapfs) [k, 2*x, (x^2 + 5)/k] representing k. They are properly equivalent (via so called R(t)-transformations) to one of the reduced forms F1 or F2. (See also W. Lang's links in A225953 and A324251, but there indefinite forms are considered.)

In order to find out which k from A343238 is represented either by form F1 or F2 the two generic multiplicative characters of Disc = -20, namely Legendre(k|p), with the odd prime p = 5 which divides Disc = -20, and Jacobi(-1|k) can be used. See Buell, pp. 51-52. They lead to the two classes of genera of Disc -20.

The present genus I, the principal one, has for odd primes p, not 5, the values Legendre(p|5) = Legendre(5|p) = +1 and Jacobi(-1|p) = Legendre(-1|p) = +1, leading for odd primes not equal to 5 to A033205. The prime 2 is not represented. The prime 5 is trivially represented. For the other genus II these two characters have values -1. There prime 2 is represented.

For composite k the prime number factorization is used, and for powers of primes the lifting theorem is employed (see, e.g., Apostol, p. 121, Theorem 5.30). The solution for prime 2 represented by form F2 = [2, 2, 3] (from the other genus II) is not liftable to powers of 2. The solution for prime 5 is also not liftable (proof by induction). The solutions of the other primes from A033205 and A106865 are uniquely liftable to powers of these primes. See A343238 for all properly represented k for Disc = -20.

For the present genus I the properly represented integers k are given by 2^a*5^b*Product_{j=1..PI} (pI_j)^(eI(j))*Product_{k=1..PII} (pII_k)^(eII(k)), with a and b from {0, 1} but if PI = PII = 0 (empty products are 1) then a = b = 0 giving a(1) = 1. The odd primes pI_j are from A033205 (== {1, 9} (mod 20)), the primes pII_k are from the odd primes of A106865 (== {3, 7}(mod 20)). The exponents of the second product are restricted: if a = 1 then PII >= 1 and Sum_{k=1..PII} eII(k) is odd. If a = 0 then PII >= 0, and if PII >= 1 then this sum is even.

Neighboring numbers k (twins) begin: [5, 6], [29, 30], [45, 46], [69, 70], [205, 206], [229, 230], [245, 246], [269, 270], [405, 406], ...

For the solutions (X, Y) of F2 = [1, 0, 5] properly representing k = a(n) see A344233.

Equivalently, number of classes of primitive positive definite binary quadratic forms of discriminant -4n as n runs through A089269: squarefree numbers congruent to 1 or 2 mod 4.

Also class number, or number of equivalence classes of positive definite binary quadratic forms of discriminant -(4n-1). - N. J. A. Sloane, May 28 2014

Since the sequences in the OEIS occasionally change their initial terms (for editorial reasons), this is an especially ill-defined sequence! - N. J. A. Sloane, Jan 01 2005

The next term, a(47), is currently unknown. - Jianing Song, Oct 07 2018

This seems to be (-1)^(n+1) times h(-4n^2) = (-1)^(n+1)*A000003(n^2), where h(k) is the class number. Verified for n <= 10^5. - Charles R Greathouse IV, Apr 28 2013

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).

Shanks's array d(n,k) n >= 1, k >= 1, is A235606.

We conjecture that the entries of the present array are all integers. More generally, we conjecture that for r = 1, 2, ... and for each n >= 1, the expansion of exp( Sum_{i >= 1} d(n,i + r)*x^i/i ) has integer coefficients. This is the case r = 1.

For the similarly defined array associated with Shanks' c(n,k) array see A262143.

Since the sequences in the OEIS occasionally change their initial terms (for editorial reasons), this is an especially ill-defined sequence! - N. J. A. Sloane, Jan 01 2005

The next term, a(47), is currently unknown. - Jianing Song, Oct 07 2018

Shanks' array c(n,k) n >= 1, k >= 0, is A235605.

We conjecture that the entries of the present array are all integers. More generally, we conjecture that for r = 0,1,2,... and for each n >= 1, the expansion of exp( Sum_{i >= 1} c(n,i + r)*x^i/i ) has integer coefficients. The case n = 1 was conjectured by Hanna in A255895.

For the similarly defined array associated with Shanks' d(n,k) array see A262144.