A181847 -id:A181847 - cp's OEIS frontend

A258171 a(n) = Sum_{d|n} phi(d)*Bell(n/d) for n>0, a(0) = 0.

0, 1, 3, 7, 19, 56, 214, 883, 4163, 21163, 116039, 678580, 4213848, 27644449, 190900217, 1382958677, 10480146333, 82864869820, 682076827740, 5832742205075, 51724158351527, 474869816158547, 4506715739125923, 44152005855084368, 445958869299027638

Offset: 0

Alois P. Heinz, May 22 2015

nonn

Dirichlet convolution of phi(n) (A000010) and the Bell numbers (A000110) (n >= 1). - Richard L. Ollerton, May 09 2021

Alois P. Heinz, Table of n, a(n) for n = 0..500

Row sums of A258170.
Similar: A078392 (numbpart), this sequence (bell), A053635 (numbcomb), A181847 and A034738 (numbcomp), A327030 (numbperm).
Cf. A000010, A000110.

with(numtheory):
A:= proc(n, k) option remember;
      add(phi(d)*k^(n/d), d=divisors(n))
    end:
T:= (n, k)-> add((-1)^(k-i)*binomial(k, i)*A(n, i), i=0..k)/k!:
a:= n-> add(T(n, k), k=0..n):
seq(a(n), n=0..30);

a[n_] := If[n == 0, 0, DivisorSum[n, EulerPhi[#] BellB[n/#] &]];
Table[a[n], {n, 0, 25}] (* Peter Luschny, Aug 27 2019 *)

a(n) = Sum_{k=0..n} A258170(n,k).

For n >= 1, a(n) = Sum_{k=1..n} Bell(gcd(n,k)). - Richard L. Ollerton, May 09 2021

New name from Peter Luschny, Aug 27 2019

A327030 a(n) = Sum_{d|n} phi(d)*(n/d)! for n > 0, a(0) = 0.

Original entry on oeis.org

0, 1, 3, 8, 28, 124, 732, 5046, 40352, 362898, 3628932, 39916810, 479002388, 6227020812, 87178296258, 1307674368272, 20922789928384, 355687428096016, 6402373706092350, 121645100408832018, 2432902008180269152, 51090942171709450128, 1124000727777647596830

Offset: 0

Peter Luschny, Aug 27 2019

nonn

Dirichlet convolution of phi(n) and n! (n >= 1). - Richard L. Ollerton, May 09 2021

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A000010, A000142, A027750.

Similar: A078392 (numbpart), A258171 (bell), A053635 (numbcomb), A181847 and A034738 (numbcomp), this sequence (numbperm).

[0] cat [&+[EulerPhi(d)*Factorial(n div d):d in Divisors(n)]:n in [1..22]]; // Marius A. Burtea, Nov 13 2019

[0] cat [&+[Factorial(Gcd(n,i)):i in [1..n]]:n in [1..22]]; // Marius A. Burtea, Nov 13 2019

with(numtheory); A327030 := n -> add(phi(d)*(n/d)!, d = divisors(n)):
seq(A327030(n), n=0..22);

a[0] = 0; a[n_] := DivisorSum[n, EulerPhi[#] * (n/#)! &]; Array[a, 23, 0] (* Amiram Eldar, May 24 2021 *)

a(n) = if (n>0, sumdiv(n, d, eulerphi(d)*(n/d)!), 0); \\ Michel Marcus, Aug 28 2019

a(n) = Sum_{i=1..n} gcd(n,i)!. - Ridouane Oudra, Nov 13 2019

cp's OEIS Frontend

A258171 a(n) = Sum_{d|n} phi(d)*Bell(n/d) for n>0, a(0) = 0.

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