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A319661
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2-rank of the class group of imaginary quadratic field with discriminant -k, k = A191483(n).
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2
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0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 2
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OFFSET
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1,9
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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. A003642).
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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(A003642(n)) = omega(A191483(n)) - 1, where omega(k) is the number of distinct prime divisors of k.
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MATHEMATICA
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PrimeNu[Select[Range[1000], Mod[#, 4] == 0 && SquareFreeQ[#/4] && Mod[#, 16] != 12&]] - 1 (* Jean-François Alcover, Aug 02 2019, after Andrew Howroyd in A191483 *)
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PROG
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(PARI) for(n=1, 1000, if(isfundamental(-n) && n%2==0, print1(omega(n) - 1, ", ")))
(Sage)
def A319661_list(len):
L = []
for n in range(2, len+1, 2):
if is_fundamental_discriminant(-n):
L.append(sloane.A001221(n) - 1)
return L
print(A319661_list(854)) # Peter Luschny, Oct 15 2018
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CROSSREFS
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Cf. A003642, A191483, A319659, A319660.
Sequence in context: A294338 A316790 A316789 * A320015 A342956 A241918
Adjacent sequences: A319658 A319659 A319660 * A319662 A319663 A319664
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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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