cp's OEIS Frontend

A319661 2-rank of the class group of imaginary quadratic field with discriminant -k, k = A191483(n).

%I A319661 #30 Feb 24 2021 08:17:30
%S A319661 0,0,1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,1,2,1,1,2,1,1,1,2,2,2,2,1,1,2,2,1,
%T A319661 2,1,1,2,1,1,1,2,3,1,1,2,1,2,1,1,2,2,2,1,1,2,2,1,2,1,1,2,1,1,2,3,1,2,
%U A319661 1,2,2,1,1,2,2,2,2,1,1,1,2,1,2,1,2,3,2
%N A319661 2-rank of the class group of imaginary quadratic field with discriminant -k, k = A191483(n).
%C A319661 The p-rank of a finite abelian group G is equal to log_p(#{x belongs to G : x^p = 1}) where p is a prime number. In this case, G is the class group of Q(sqrt(-k)), and #{x belongs to G : x^p = 1} is the number of genera of Q(sqrt(-k)) (cf. A003642).
%H A319661 Rick L. Shepherd, <a href="https://libres.uncg.edu/ir/uncg/listing.aspx?id=15057">Binary quadratic forms and genus theory</a>, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
%F A319661 a(n) = log_2(A003642(n)) = omega(A191483(n)) - 1, where omega(k) is the number of distinct prime divisors of k.
%t A319661 PrimeNu[Select[Range[1000], Mod[#, 4] == 0 && SquareFreeQ[#/4] && Mod[#, 16] != 12&]] - 1 (* _Jean-François Alcover_, Aug 02 2019, after _Andrew Howroyd_ in A191483 *)
%o A319661 (PARI) for(n=1, 1000, if(isfundamental(-n) && n%2==0, print1(omega(n) - 1, ", ")))
%o A319661 (Sage)
%o A319661 def A319661_list(len):
%o A319661     L = []
%o A319661     for n in range(2, len+1, 2):
%o A319661         if is_fundamental_discriminant(-n):
%o A319661             L.append(sloane.A001221(n) - 1)
%o A319661     return L
%o A319661 print(A319661_list(854)) # _Peter Luschny_, Oct 15 2018
%Y A319661 Cf. A003642, A191483, A319659, A319660.
%K A319661 nonn
%O A319661 1,9
%A A319661 _Jianing Song_, Sep 25 2018