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

Author

Alois P. Heinz, May 22 2015

Keywords

• nonn

Comments

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

Links

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

Crossrefs

Row sums of A258170.

Similar: A078392 (numbpart), this sequence (bell), A053635 (numbcomb), A181847 and A034738 (numbcomp), A327030 (numbperm).

Cf. A000010, A000110.

Programs

Maple

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);

Mathematica

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

Formula

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

Extensions

New name from Peter Luschny, Aug 27 2019