A003643 Number of genera of Q(sqrt(-n)), n squarefree. (Formerly M0194)

1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 2, 1, 4, 2, 1, 2, 2, 4, 1, 4, 2, 2, 2, 2, 2, 2, 4, 1, 2, 1, 2, 2, 2, 4, 2, 1, 2, 2, 4, 4, 1, 4, 4, 1, 2, 2, 4, 4, 1, 2, 1, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 1, 8, 2, 1, 2, 4, 2, 2, 4, 2, 2, 2, 2, 2, 1, 4, 4, 1, 4, 2, 2, 4, 1, 4, 2, 2, 4, 2, 2, 1, 4, 2, 2, 2, 2, 4, 1, 8, 2, 1, 4, 2

Offset

1,4

References

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Index entries for sequences related to quadratic fields

Formula

a(n) = 2^(omega(A033197(n)) - 1). - Andrew Howroyd, Jul 24 2018

Let k = A005117(n) be the n-th squarefree number, then a(n) = 2^omega(k) if k == 1 (mod 4) and 2^(omega(k) - 1) otherwise. - Jianing Song, Jul 25 2018

Mathematica

Function[If[Mod[#, 4] == 1, 2^PrimeOmega[#], 2^(PrimeOmega[#] - 1)]] /@ Select[Range[200], SquareFreeQ] (* Jean-François Alcover, Sep 04 2019 *)

Prog

(PARI) for(n=1, 200, if(issquarefree(n), print1(2^(omega(n*if((-n)%4>1, 4, 1)) - 1), ", "))) \\ Andrew Howroyd, Jul 24 2018

Crossrefs

Cf. A000924, A001221 (omega), A005117, A033197, A003640.

Sequence in context: A191971 A156051 A091267 * A092788 A058062 A102820

Adjacent sequences: A003640 A003641 A003642 * A003644 A003645 A003646

Keyword

nonn,nice

Author

N. J. A. Sloane, Mira Bernstein

Status

approved