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A317991 2-rank of the class group of real quadratic field with discriminant A003658(n), n >= 2. 2


%S 0,0,1,0,0,1,1,1,0,1,0,1,0,1,0,1,1,2,0,1,1,0,1,1,1,1,0,1,1,0,0,1,2,0,

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

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

%N 2-rank of the class group of real quadratic field with discriminant A003658(n), n >= 2.

%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 narrow class group of Q[sqrt(k)] or the form class group of indefinite binary quadratic forms with discriminant k, and #{x belongs to G : x^p = 1} is the number of genera of Q[sqrt(k)] (cf. A317989).

%C This is the analog of A319659 for real quadratic fields.

%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) = omega(A003658(n)) - 1 = log_2(A317989(n)), where omega(k) is the number of distinct prime divisors of k.

%t PrimeNu[Select[Range[2, 300], NumberFieldDiscriminant[Sqrt[#]] == #&]] - 1 (* _Jean-Fran├žois Alcover_, Jul 25 2019 *)

%o (PARI) for(n=2, 1000, if(isfundamental(n), print1(omega(n) - 1, ", ")))

%Y Cf. A003658, A087048, A317989, A317992, A319659.

%K nonn

%O 2,18

%A _Jianing Song_, Oct 03 2018

%E Offset corrected by _Jianing Song_, Mar 31 2019

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Last modified November 22 03:34 EST 2019. Contains 329385 sequences. (Running on oeis4.)