A208769 - OEIS

%I #23 Mar 21 2024 14:25:07

%S 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,

%T -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,

%U 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

%N Dirichlet inverse of the finite group count (A000001).

%H Antti Karttunen, <a href="/A208769/b208769.txt">Table of n, a(n) for n = 1..2047</a> (computed from the b-file of A000001; a(1024) corrected by Andrey Zabolotskiy)

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_convolution#Dirichlet_inverse">Dirichlet convolution (Dirichlet inverse)</a>

%F a(1) = 1; for n > 1, a(n) = -Sum_{d|n, d<n} A000001(n/d)*a(d). - _Antti Karttunen_, Jun 13 2018

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

%o (PARI)

%o v000001 = readvec("b000001_to.txt"); \\ Prepared with gawk ' { print $2 } ' from the b-file of A000001.

%o A000001(n) = v000001[1+n];

%o A208769(n) = if(1==n,1,-sumdiv(n,d,if(d<n,A000001(n/d)*A208769(d),0))); \\ _Antti Karttunen_, Jun 13 2018, after Mathematica-code

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

%K sign,hard

%O 1,8

%A _Ben Branman_, Mar 01 2012

%E More terms from _Antti Karttunen_, Jun 13 2018