A244891 - OEIS

%I #10 Feb 24 2024 11:08:04

%S 1,2,1,3,4,1,1,6,7,2,9,2,2,12,13,3,16,3,18,19,3,69,22,4,4,5,31,28,5,5,

%T 39,38,6,44,7,49,7,50,47,8,54,8,61,10,9,71,74,231,66,30,12,79,14,84,

%U 39,85,12,105,14,16,104,13,114,111,19,14,136,22,116,17,19,405,151,20,164,133

%N Class number of maximal order O_1 in a totally definite quaternion Q(sqrt(p))-algebra D_{oo_1, oo_2}, where p = n-th prime.

%H J. Xue, T.-C. Yang, C.-F. Yu, <a href="http://arxiv.org/abs/1404.2978">Supersingular abelian surfaces and Eichler class number formula</a>, arXiv preprint arXiv:1404.2978, 2014. See Table 1.

%o (Sage)

%o def a(n):

%o     if n < 4: return [1,2,1][n-1]

%o     p = Primes()[n-1]

%o     F = NumberField(x^2 - p, names='a')

%o     hKi = [F.extension(x^2+i+1, names='b').class_number() for i in range(3)]

%o     ans = F.class_number()*F.zeta_function(100)(-1)/2 + hKi[2]/3

%o     if p%4 == 1: ans += hKi[0]/4

%o     else: ans += (3 + 5*(2 - legendre_symbol(2,p)))*hKi[0]/8 + hKi[1]/4

%o     return round(ans.real_part())  # _Robin Visser_, Feb 23 2024

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Jul 13 2014

%E More terms from _Robin Visser_, Feb 23 2024