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A319662
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2-rank of the class group of Q(sqrt(-k)), k squarefree.
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2
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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
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OFFSET
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1,14
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COMMENTS
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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).
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LINKS
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Table of n, a(n) for n=1..87.
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
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FORMULA
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a(n) = log_2(A003643(n)) = omega(A005117(n)) - 1, where omega(k) is the number of distinct prime divisors of k.
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MATHEMATICA
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PrimeNu[#*If[Mod[-#, 4]>1, 4, 1]] - 1& /@ Select[Range[200], SquareFreeQ] (* Jean-François Alcover, Aug 02 2019 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Jianing Song, Sep 25 2018
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STATUS
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approved
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