cp's OEIS Frontend

A317989 Number of genera of real quadratic field with discriminant A003658(n), n >= 2.

%I A317989 #25 Mar 16 2020 13:06:43
%S A317989 1,1,2,1,1,2,2,2,1,2,1,2,1,2,1,2,2,4,1,2,2,1,2,2,2,2,1,2,2,1,1,2,4,1,
%T A317989 1,4,2,2,2,2,1,4,2,2,1,2,4,1,2,4,4,2,1,2,1,2,2,2,1,1,2,4,2,2,2,2,4,2,
%U A317989 1,2,1,2,2,1,2,2,2,1,4,2,2,1,4,1,4,1,2
%N A317989 Number of genera of real quadratic field with discriminant A003658(n), n >= 2.
%C A317989 The number of genera of a quadratic field is equal to the number of elements x in the class group such that x^2 = e where e is the identity.
%C A317989 This is the analog of A003640 for real quadratic fields. Note that for this case "the class group" refers to the narrow class group, or the form class group of indefinite binary quadratic forms with discriminant k.
%H A317989 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 A317989 a(n) = 2^(omega(A003658(n)-1)) = 2^A317991(n), where omega(k) is the number of distinct prime divisors of k.
%t A317989 2^(PrimeNu[Select[Range[2, 300], NumberFieldDiscriminant[Sqrt[#]]==#&]] - 1) (* _Jean-François Alcover_, Jul 25 2019 *)
%o A317989 (PARI) for(n=2, 1000, if(isfundamental(n), print1(2^(omega(n) - 1), ", ")))
%o A317989 (PARI) for(n=2, 1000, if(isfundamental(n), print1(2^#select(t->t%2==0, quadclassunit(n).cyc), ", ")))
%o A317989 (Sage)
%o A317989 def A317989_list(len):
%o A317989     L = (sloane.A001221(n) for n in (1..len) if is_fundamental_discriminant(n))
%o A317989     return [2^(l-1) for l in L]
%o A317989 A317989_list(290) # _Peter Luschny_, Oct 15 2018
%Y A317989 Cf. A003640, A003658, A087048, A317990, A317991.
%K A317989 nonn
%O A317989 2,3
%A A317989 _Jianing Song_, Oct 03 2018
%E A317989 Offset corrected by _Jianing Song_, Mar 31 2019