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A363148
a(n) gives the number of equivalence classes of quaternary quadratic forms of discriminant A363147(n) not representing 2.
2
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
1,3
COMMENTS
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.
LINKS
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.
EXAMPLE
a(5) = 1 as there is only one equivalence class of quaternary quadratic form of discriminant A363147(5) = 277 not representing 2 (see A307250).
PROG
(Sage)
bound = 100
P = Primes()
p = 2
for i in range(bound):
p = P.next(p)
if p % 4 == 1:
K1.<a> = NumberField(x^2 - p)
K2.<b> = NumberField(x^2 + p)
K3.<c> = 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)
CROSSREFS
Sequence in context: A287731 A289816 A374412 * A054482 A208234 A092543
KEYWORD
nonn
AUTHOR
Andy Huchala, May 17 2023
STATUS
approved