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a(n) = Sum_{d|n} phi(d)*(n/d)! for n > 0, a(0) = 0.
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%I #28 Sep 08 2022 08:46:24

%S 0,1,3,8,28,124,732,5046,40352,362898,3628932,39916810,479002388,

%T 6227020812,87178296258,1307674368272,20922789928384,355687428096016,

%U 6402373706092350,121645100408832018,2432902008180269152,51090942171709450128,1124000727777647596830

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

%C Dirichlet convolution of phi(n) and n! (n >= 1). - _Richard L. Ollerton_, May 09 2021

%H Seiichi Manyama, <a href="/A327030/b327030.txt">Table of n, a(n) for n = 0..449</a>

%F a(n) = Sum_{i=1..n} gcd(n,i)!. - _Ridouane Oudra_, Nov 13 2019

%p with(numtheory); A327030 := n -> add(phi(d)*(n/d)!, d = divisors(n)):

%p seq(A327030(n), n=0..22);

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

%o (PARI) a(n) = if (n>0, sumdiv(n, d, eulerphi(d)*(n/d)!), 0); \\ _Michel Marcus_, Aug 28 2019

%o (Magma) [0] cat [&+[EulerPhi(d)*Factorial(n div d):d in Divisors(n)]:n in [1..22]]; // _Marius A. Burtea_, Nov 13 2019

%o (Magma) [0] cat [&+[Factorial(Gcd(n,i)):i in [1..n]]:n in [1..22]]; // _Marius A. Burtea_, Nov 13 2019

%Y Cf. A000010, A000142, A027750.

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

%K nonn

%O 0,3

%A _Peter Luschny_, Aug 27 2019