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a(n) = Sum_{d|n} phi(d)*n^(n/d).
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%I #36 Sep 08 2022 08:46:05

%S 0,1,6,33,280,3145,46956,823585,16781472,387422001,10000100440,

%T 285311670721,8916103479504,302875106592409,11112006930972780,

%U 437893890382391745,18446744078004651136,827240261886336764449,39346408075494964903956,1978419655660313589124321

%N a(n) = Sum_{d|n} phi(d)*n^(n/d).

%H Alois P. Heinz, <a href="/A228640/b228640.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = Sum_{k=1..n} n^gcd(k,n) = n * A056665(n). - _Seiichi Manyama_, Mar 10 2021

%F a(n) = Sum_{k=1..n} n^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - _Richard L. Ollerton_, May 07 2021

%p with(numtheory):

%p a:= n-> add(phi(d)*n^(n/d), d=divisors(n)):

%p seq(a(n), n=0..20);

%t a[0] = 0; a[n_] := DivisorSum[n, EulerPhi[#]*n^(n/#)&]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 21 2017 *)

%o (Python)

%o from sympy import totient, divisors

%o def A228640(n):

%o return sum(totient(d)*n**(n//d) for d in divisors(n,generator=True)) # _Chai Wah Wu_, Feb 15 2020

%o (PARI) a(n) = if (n, sumdiv(n, d, eulerphi(d)*n^(n/d)), 0); \\ _Michel Marcus_, Feb 15 2020; corrected Jun 13 2022

%o (PARI) a(n) = sum(k=1, n, n^gcd(k, n)); \\ _Seiichi Manyama_, Mar 10 2021

%o (Magma) [0] cat [&+[EulerPhi(d)*n^(n div d): d in Divisors(n)]:n in [1..20]]; // _Marius A. Burtea_, Feb 15 2020

%Y Main diagonal of A054618, A054619, A185651.

%Y Cf. A000010, A056665.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 28 2013