A349612 Dirichlet convolution of A342001 [{arithmetic derivative of n}/A003557(n)] with A325126 [Dirichlet inverse of rad(n)].

0, 1, 1, 0, 1, 0, 1, 1, -1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -3, 0, 3, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -5, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, -5, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset

1,25

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = Sum_{d|n} A342001(d) * A325126(n/d).

If p prime, a(p) = 1. - _Bernard Schott_, Nov 28 2021

Dirichlet g.f.: Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)). - _Sebastian Karlsson_, May 05 2022

Mathematica

f[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f @@@ FactorInteger[n]; f1[p_, e_] := p^(e-1); s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := -p*(1 - p)^(e - 1); s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, d[#]*s2[n/#]/s1[#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 23 2021 *)

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
memoA325126 = Map();
A325126(n) = if(1==n, 1, my(v); if(mapisdefined(memoA325126, n, &v), v, v = -sumdiv(n, d, if(d<n, A007947(n/d)*A325126(d), 0)); mapput(memoA325126, n, v); (v)));
A349612(n) = sumdiv(n, d, A342001(d)*A325126(n/d));

Crossrefs

Cf. A003415, A003557, A007947, A342001, A325126.

Cf. also A349394, A349396, A349618.

Sequence in context: A094901 A030220 A219240 * A277080 A055240 A174559

Adjacent sequences: A349609 A349610 A349611 * A349613 A349614 A349615

Keyword

sign

Author

_Antti Karttunen_, Nov 23 2021

Status

approved