A347135 - OEIS

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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

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OFFSET

1,4

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

FORMULA

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

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

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 *)

PROG

(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));

CROSSREFS

Cf. A001221, A001615, A069359, A322577.

Cf. also A347132, A347133, A347134.

Sequence in context: A113261 A063004 A351573 * A146993 A343223 A104572

Adjacent sequences: A347132 A347133 A347134 * A347136 A347137 A347138

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 23 2021

STATUS

approved