A304820 A co-delta function for non-perfect powers. Dirichlet inverse of A304819.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0

Offset

1,36

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = Sum_{d|n} A304779(d) * mu(n/d), where A304779 is the Dirichlet inverse of A304653.

Mathematica

a[n_]:=a[n]=If[n==1, 1, -Sum[(-1)^PrimeOmega[d]*a[n/d], {d, Select[Rest[Divisors[n]], GCD@@FactorInteger[#][[All, 2]]==1&]}]];
Table[Sum[a[d]*MoebiusMu[n/d], {d, Divisors[n]}], {n, 100}]

Prog

(PARI)
A304819(n) = sumdiv(n, d, if(!ispower(d), (-1)^bigomega(d), 0));
A304820(n) = if(1==n, 1, -sumdiv(n, d, if(d<n, A304819(n/d)*A304820(d), 0))); \\ Antti Karttunen, Jul 29 2018
(PARI)
A304779(n) = if(1==n, 1, -sumdiv(n, d, if((d>1)&&!ispower(d), ((-1)^bigomega(d))*A304779(n/d), 0)));
A304820(n) = sumdiv(n, d, moebius(n/d)*A304779(d)); \\ Antti Karttunen, Jul 29 2018

Crossrefs

Positions of nonzero entries appear to be A126706.

Cf. A000005, A000007, A001222, A001597, A005117, A007916, A008683, A008836, A304362, A304653, A304779, A304817, A304819.

Sequence in context: A256574 A369427 A304819 * A162641 A333487 A348380

Adjacent sequences: A304817 A304818 A304819 * A304821 A304822 A304823

Keyword

nonn

Author

Gus Wiseman, May 19 2018

Extensions

More terms from Antti Karttunen, Jul 29 2018

Status

approved