A307250 -id:A307250 - cp's OEIS frontend

bound = 100 P = Primes() p = 2 for i in range(bound): p = P.next(p) if p % 4 == 1: K1. = NumberField(x^2 - p) K2. = NumberField(x^2 + p) K3. = NumberField(x^2 + 3*p) zeta = K1.zeta_function() h2 = len(K2.class_group()) h3 = len(K3.class_group()) H_plus = int(abs(.49+1/2*zeta(-1)+1/8 * h2 + 1/6*h3)) H = (H_plus+int((p + 19)/24))/2 if H_plus-H>0: print(p)

A363148 a(n) gives the number of equivalence classes of quaternary quadratic forms of discriminant A363147(n) not representing 2.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 4, 1, 2, 2, 1, 1, 4, 6, 2, 6, 5, 7, 1, 1, 7, 4, 2, 9, 10, 7, 13, 5, 8, 11, 3, 5, 15, 3, 5, 7, 6, 8, 14, 20, 3, 4, 17, 6, 9, 8, 15, 10, 19, 20, 26, 7, 20, 20, 12, 34, 7, 13, 32, 26, 10, 16, 16, 23, 11, 17, 41, 37, 11, 28, 46, 20, 28, 14, 17

Offset: 1

Andy Huchala, May 17 2023

nonn

Conjecture: a(n) ~ c * A363147(n) ^ d where d is a constant which is roughly 1.51 and c is one of four constants, depending on the value of A363147(n) mod 24. See plots in files.

			a(5) = 1 as there is only one equivalence class of quaternary quadratic form of discriminant A363147(5) = 277 not representing 2 (see A307250).

Andy Huchala, Table of n, a(n) for n = 1..20000
Andy Huchala, Growth of A363147(n) vs a(n)
F. Hirzebruch, Modulflächen und Modulkurven zur symmetrischen Hilbertschen Modulgruppe, Annales scientifiques de l’É.N.S. 4e série, tome 11, no 1 (1978), p. 101-165. See page 135.
Jürg Kramer, On the linear independence of certain theta-series, Mathematische Annalen 281.2 (1988): 219-228. See page 226.

Cf. A307250, A363147.

bound = 100
P = Primes()
p = 2
for i in range(bound):
    p = P.next(p)
    if p % 4 == 1:
        K1. = NumberField(x^2 - p)
        K2. = NumberField(x^2 + p)
        K3. = NumberField(x^2 + 3*p)
        zeta = K1.zeta_function()
        h2 = len(K2.class_group())
        h3 = len(K3.class_group())
        H_plus = int(abs(.49+1/2*zeta(-1)+1/8 * h2 + 1/6*h3))
        H = (H_plus+int((p + 19)/24))/2
        if H_plus-H>0:
            print(H_plus-H)

cp's OEIS Frontend

A363147 Primes q == 1 (mod 4) such that there is at least one equivalence class of quaternary quadratic forms of discriminant q not representing 2.

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