A349434 Dirichlet convolution of A129283 (n + its arithmetic derivative) with A349337 (Dirichlet inverse of A230593).

1, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, -2, 0, 0, 0, 6, 0, -3, 0, -2, 0, 0, 0, 0, 5, 0, 6, -2, 0, 0, 0, 10, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, -2, -3, 0, 0, -6, 7, -5, 0, -2, 0, -3, 0, 0, 0, 0, 0, 4, 0, 0, -3, 22, 0, 0, 0, -2, 0, 0, 0, -5, 0, 0, -5, -2, 0, 0, 0, -6, 21, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, -2, 0, 0, 0, -4, 0, -7, -3, 7

Offset: 1

Author

Antti Karttunen, Nov 17 2021

Keywords

• sign

Comments

Dirichlet convolution of this sequence with A349338 is A348976.

Links

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

Crossrefs

Cf. A003415, A129283, A230593, A349337, A349435 (Dirichlet inverse), A349436 (sum with it).

Cf. also A348976, A349338.

Programs

Mathematica

s[n_] := n * DivisorSum[n, 1/# &, !CompositeQ[#] &]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; f[p_, e_] := e/p; d[1] = 1; d[n_] := n*(1 + Plus @@ f @@@ FactorInteger[n]); a[n_] := DivisorSum[n, sinv[#] * d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

PARI

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A129283(n) = (n+A003415(n));
A230593(n) = sumdiv(n, d, ((1==d)||isprime(d))*(n/d));
v349337 = DirInverseCorrect(vector(up_to,n,A230593(n)));
A349337(n) = v349337[n];
A349434(n) = sumdiv(n,d,A129283(n/d)*A349337(d));

Formula

a(n) = Sum_{d|n} A129283(n/d) * A349337(d).