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A319659
2-rank of the class group of imaginary quadratic field with discriminant -k, k = A003657(n).
4
0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 2, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 1, 1, 0, 0, 2, 2, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2, 0, 2, 1, 0, 2, 2
OFFSET
1,27
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. A003640).
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(A003640(n)) = omega(A003657(n)) - 1, where omega(k) is the number of distinct prime divisors of k.
PROG
(PARI) for(n=1, 1000, if(isfundamental(-n), print1(omega(n) - 1, ", ")))
CROSSREFS
Real discriminant case: A317991.
Sequence in context: A116663 A258940 A340607 * A050372 A037802 A037879
KEYWORD
nonn
AUTHOR
Jianing Song, Sep 25 2018
STATUS
approved