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

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

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• nice

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

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