A349435 Dirichlet convolution of A230593 with A347084, which is Dirichlet inverse of {n + its arithmetic derivative}.

1, 0, 0, -2, 0, 0, 0, -2, -3, 0, 0, 2, 0, 0, 0, -2, 0, 3, 0, 2, 0, 0, 0, 0, -5, 0, -6, 2, 0, 0, 0, -2, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, -2, -7, 5, 0, 2, 0, 3, 0, 0, 0, 0, 0, -4, 0, 0, 3, -2, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 5, 2, 0, 0, 0, -2, -12, 0, 0, -4, 0, 0, 0, 0, 0, -6, 0, 2, 0, 0, 0, -4, 0, 7, 3

Offset: 1

Author

Antti Karttunen, Nov 17 2021

Keywords

• sign

Comments

Dirichlet convolution of this sequence with A348976 is A349338.

The positions of records start as: 1, 12, 18, 36, 100, 108, 196, 225, 324, 441, 500, 1125, 1372, 2500, 5000, 5324, 8575, 8788, 9604, 12500, 19652, etc.

Links

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

Crossrefs

Cf. A003415, A129283, A230593, A347084, A349434 (Dirichlet inverse), A349436 (sum with it).

Cf. also A348976, A349338.

Programs

Mathematica

s[n_] := n * DivisorSum[n, 1/# &, !CompositeQ[#] &]; f[p_, e_] := e/p; d[1] = 1; d[n_] := n*(1 + Plus @@ f @@@ FactorInteger[n]); dinv[1] = 1; dinv[n_] := dinv[n] = -DivisorSum[n, dinv[#] * d[n/#] &, # < n &]; a[n_] := DivisorSum[n, s[#] * dinv[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]));
A230593(n) = sumdiv(n, d, ((1==d)||isprime(d))*(n/d));
v347084 = DirInverseCorrect(vector(up_to,n,n+A003415(n)));
A347084(n) = v347084[n];
A349435(n) = sumdiv(n,d,A230593(n/d)*A347084(d));

Formula

a(n) = Sum_{d|n} A230593(n/d) * A347084(d).