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

Author

Gus Wiseman, May 19 2018

Keywords

• nonn

Links

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

Crossrefs

Positions of nonzero entries appear to be A126706.

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

Programs

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}]

PARI

A304819(n) = sumdiv(n,d,if(!ispower(d),(-1)^bigomega(d),0));
A304820(n) = if(1==n,1,-sumdiv(n,d,if(dA304819(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

Formula

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

Extensions

More terms from Antti Karttunen, Jul 29 2018