A319660 - OEIS

%I #18 Feb 24 2021 08:17:17

%S 0,0,0,1,0,0,0,1,1,0,0,1,1,0,0,0,0,0,1,1,1,0,0,1,1,1,1,0,0,0,1,0,1,1,

%T 0,0,0,1,1,0,2,0,1,0,1,1,0,0,2,1,0,1,0,2,1,0,1,0,0,1,1,1,1,1,0,0,1,1,

%U 1,0,1,1,0,1,0,0,1,0,0,1,1,2,1,1,1,1,0

%N 2-rank of the class group of imaginary quadratic field with discriminant -k, k = A039957(n).

%C 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. A003641).

%H 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 a(n) = log_2(A003641(n)) = omega(A039957(n)) - 1, where omega(k) is the number of distinct prime divisors of k.

%o (PARI) for(n=1, 1000, if(n%4==3 && issquarefree(n), print1(omega(n) - 1, ", ")))

%Y Cf. A003641, A039957, A319659, A319661.

%K nonn

%O 1,41

%A _Jianing Song_, Sep 25 2018