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 A319662 2-rank of the class group of Q(sqrt(-k)), k squarefree. 2
 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 2, 0, 2, 1, 1, 1, 1, 1, 1, 2, 0, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 2, 2, 0, 2, 2, 0, 1, 1, 2, 2, 0, 1, 0, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 0, 3, 1, 0, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 0, 2, 2, 0, 2, 1, 1, 2, 0, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,14 COMMENTS 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. A003643). LINKS Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013. FORMULA a(n) = log_2(A003643(n)) = omega(A005117(n)) - 1, where omega(k) is the number of distinct prime divisors of k. MATHEMATICA PrimeNu[#*If[Mod[-#, 4]>1, 4, 1]] - 1& /@ Select[Range[200], SquareFreeQ] (* Jean-François Alcover, Aug 02 2019 *) PROG (PARI) for(n=1, 200, if(issquarefree(n), print1(omega(n*if((-n)%4>1, 4, 1)) - 1, ", "))) (Sage) def A319662_list(len): L = [] for n in (1..len): if is_squarefree(n): if (-n) % 4 > 1: n <<= 2 L.append(sloane.A001221(n) - 1) return L print(A319662_list(141)) # Peter Luschny, Oct 15 2018 CROSSREFS Cf. A000924, A003643, A005117, A033197, A319659. Real discriminant case: A317992. Sequence in context: A113446 A121450 A143110 * A109294 A132966 A037897 Adjacent sequences: A319659 A319660 A319661 * A319663 A319664 A319665 KEYWORD nonn AUTHOR Jianing Song, Sep 25 2018 STATUS approved

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Last modified March 30 12:42 EDT 2023. Contains 361618 sequences. (Running on oeis4.)