A347131 a(n) = Sum_{d|n} phi(n/d) * A003415(d), where A003415 is the arithmetic derivative and phi is Euler totient function

0, 1, 1, 5, 1, 8, 1, 18, 8, 12, 1, 33, 1, 16, 14, 56, 1, 45, 1, 53, 18, 24, 1, 110, 14, 28, 45, 73, 1, 87, 1, 160, 26, 36, 22, 169, 1, 40, 30, 182, 1, 119, 1, 113, 93, 48, 1, 328, 20, 107, 38, 133, 1, 216, 30, 254, 42, 60, 1, 337, 1, 64, 125, 432, 34, 183, 1, 173, 50, 183, 1, 538, 1, 76, 135, 193, 34, 215, 1, 552, 216

Offset

1,4

Comments

Dirichlet convolution of A000010 with A003415.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Formula

a(n) = Sum_{d|n} A000010(n/d) * A003415(d).

a(n) = Sum_{d|n} A008683(n/d) * A347130(d).

a(n) = Sum_{k=1..n} A003415(gcd(n,k)). - Antti Karttunen, Sep 02 2021

Mathematica

f[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, d[#] * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Sep 03 2021 *)

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A347131(n) = sumdiv(n, d, A003415(n/d)*eulerphi(d));
(PARI) A347131(n) = sum(k=1, n, A003415(gcd(n, k))); \\ (Slow) - Antti Karttunen, Sep 02 2021

Crossrefs

Möbius transform of A347130.

Cf. A000010, A003415.

Cf. also A300251, A305809, A322577, A347132, A347133, A347134, A347235.

Sequence in context: A363514 A347955 A349133 * A350515 A073116 A201525

Adjacent sequences: A347128 A347129 A347130 * A347132 A347133 A347134

Keyword

nonn

Author

Antti Karttunen, Aug 23 2021

Status

approved