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

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

Author

Peter Luschny, Aug 27 2019

Keywords

• nonn

Comments

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

Links

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

Crossrefs

Cf. A000010, A000142, A027750.

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

Programs

Magma

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

Magma

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

Maple

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

Mathematica

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

PARI

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

Formula

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