A208769 Dirichlet inverse of the finite group count (A000001).

1, -1, -1, -1, -1, 0, -1, -2, -1, 0, -1, 0, -1, 0, 1, -5, -1, 0, -1, 0, 0, 0, -1, -1, -1, 0, -2, 1, -1, 0, -1, -23, 1, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 1, 1, 0, -1, -8, -1, 0, 1, 0, -1, -1, 0, -1, 0, 0, -1, 0, -1, 0, 1, -159, 1, 0, -1, 0, 1, 0, -1, -6, -1, 0, 0, 1, 1, 0, -1, -10, -6, 0, -1, 1, 1, 0, 1, 0, -1, 0, 1, 1, 0, 0, 1, -60, -1, 0, 1, -2, -1, 0, -1, 0

Offset: 1

Author

Ben Branman, Mar 01 2012

Keywords

• sign
• hard

Links

• Antti Karttunen, Table of n, a(n) for n = 1..2047 (computed from the b-file of A000001; a(1024) corrected by Andrey Zabolotskiy)
• Wikipedia, Dirichlet convolution (Dirichlet inverse)

Crossrefs

Cf. A129667 (abelian version), A000688, A000001, A185291.

Programs

Mathematica

a[1] = 1; a[n_] := a[n] = -Sum[FiniteGroupCount[n/k] a[k], {k, Drop[Divisors[n], -1]}]; Table[a[n], {n, 100}]

PARI

v000001 = readvec("b000001_to.txt"); \\ Prepared with gawk ' { print $2 } ' from the b-file of A000001.
A000001(n) = v000001[1+n];
A208769(n) = if(1==n,1,-sumdiv(n,d,if(dA000001(n/d)*A208769(d),0))); \\ Antti Karttunen, Jun 13 2018, after Mathematica-code

Formula

a(1) = 1; for n > 1, a(n) = -Sum_{d|n, dA000001(n/d)*a(d). - Antti Karttunen, Jun 13 2018

Extensions

More terms from Antti Karttunen, Jun 13 2018