A347135 a(n) = Sum_{d|n} A001615(n/d) * A069359(d).

0, 1, 1, 5, 1, 12, 1, 16, 7, 16, 1, 51, 1, 20, 18, 44, 1, 68, 1, 71, 22, 28, 1, 156, 11, 32, 33, 91, 1, 167, 1, 112, 30, 40, 26, 277, 1, 44, 34, 220, 1, 215, 1, 131, 110, 52, 1, 420, 15, 140, 42, 151, 1, 300, 34, 284, 46, 64, 1, 673, 1, 68, 138, 272, 38, 311, 1, 191, 54, 295, 1, 836, 1, 80, 162, 211, 38, 359, 1, 596

Offset: 1

Author

Antti Karttunen, Aug 23 2021

Keywords

• nonn

Comments

Dirichlet convolution of A001615 (Dedekind psi function) with A069359.

Dirichlet convolution of A001221 (omega, number of distinct prime factors of n) with A322577.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Wikipedia, Dirichlet convolution

Crossrefs

Cf. A001221, A001615, A069359, A322577.

Cf. also A347132, A347133, A347134.

Programs

Mathematica

Table[DivisorSum[n,PrimeNu[n/#]*Sum[DirichletConvolve[j,MoebiusMu[j]^2,j,#/d] EulerPhi[d],{d,Divisors[#]}]&],{n,80}] (* Giorgos Kalogeropoulos, Oct 28 2021 *)

PARI

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A347135(n) = sumdiv(n,d,A001615(n/d)*A069359(d));

Formula

a(n) = Sum_{d|n} A001615(n/d) * A069359(d).

a(n) = Sum_{d|n} A001221(n/d) * A322577(d).