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%I #13 Mar 31 2019 04:08:55
%S 1,2,1,2,2,2,2,1,2,4,1,2,2,2,2,2,1,4,2,2,2,4,1,2,4,1,4,2,2,2,4,1,4,2,
%T 2,2,1,2,2,4,2,2,4,2,1,2,2,4,2,2,2,2,2,4,1,4,2,2,4,1,1,4,2,4,2,2,1,4,
%U 4,1,4,4,2,4,2,4,2,2,4,2,2,2,1,4,2,2,2
%N Number of genera of real quadratic field Q(sqrt(k)), k squarefree > 1.
%C 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 This is the analog of A003643 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 forms with discriminant k.
%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) = 2^(omega(A005117(n)-1)) = 2^A317992(n), where omega(k) is the number of distinct prime divisors of k.
%o (PARI) for(n=2, 200, if(issquarefree(n), print1(2^(omega(n*if(n%4>1, 4, 1)) - 1), ", ")))
%Y Cf. A003643, A005117, A087048, A317989, A317992.
%K nonn
%O 2,2
%A _Jianing Song_, Oct 03 2018
%E Offset corrected by _Jianing Song_, Mar 31 2019
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