A304779 The "rootless" zeta function. Dirichlet inverse of the function defined by r(n) = (-1)^Omega(n) if n is 1 or not a perfect power and r(n) = 0 otherwise.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 7, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 2, 2, 5, 1, 1, 1, 2, 1

Offset: 1

Author

Gus Wiseman, May 18 2018

Keywords

• nonn

Comments

Omega(n) = A001222(n) is the number of prime factors of n counted with multiplicity.

First occurrence of k: 1, 12, 48, 60, 36, 3072, 72, 420, 240, 786432, 3145728, 144, 216, ..., . - Robert G. Wilson v, Jul 22 2018

Records: 1, 2, 5, 7, 12, 13, 15, 18, 26, 37, 38, 57, 60, 67, 81, 96, 142, 165, 199, 221, 234, ..., . - Robert G. Wilson v, Jul 22 2018

Links

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

Crossrefs

Positions of entries greater than 1 appear to be A126706.

Cf. A000005, A000012, A000961, A001221, A001222, A001597, A005117, A007916, A008683, A008966, A091050, A303554, A304326, A304362, 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&]}]];
Array[a,100]

PARI

A304779(n) = if(1==n,1,-sumdiv(n,d,if((d>1)&&!ispower(d),((-1)^bigomega(d))*A304779(n/d),0))); \\ Antti Karttunen, Jul 22 2018

Formula

a(1) = 1 and a(n > 1) = -Sum_{d|n, d not a perfect power} (-1)^Omega(d) * a(n/d).

Extensions

More terms from Antti Karttunen, Jul 22 2018