A309369 - OEIS

%I #15 Mar 13 2021 10:52:16

%S 1,2,3,4,5,8,7,10,15,22,11,34,13,44,105,42,17,116,19,314,357,112,23,

%T 426,1045,158,747,1474,29,5290,31,594,3069,274,24185,6082,37,344,9945,

%U 67922,41,63542,43,12170,303225,508,47,74834,279979,1050022,135201,29098,53,309872,4294345

%N a(n) = Sum_{d|n} phi(n/d)^d, where phi = Euler totient function (A000010).

%H Seiichi Manyama, <a href="/A309369/b309369.txt">Table of n, a(n) for n = 1..5000</a>

%F G.f.: Sum_{k>=1} phi(k)*x^k/(1 - phi(k)*x^k).

%F L.g.f.: -log(Product_{k>=1} (1 - phi(k)*x^k)^(1/k)).

%F a(p) = p for p prime.

%F a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^(gcd(k, n) - 1). - _Seiichi Manyama_, Mar 13 2021

%t Table[Sum[EulerPhi[n/d]^d, {d, Divisors[n]}], {n, 1, 55}]

%t nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 - EulerPhi[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t nmax = 55; CoefficientList[Series[-Log[Product[(1 - EulerPhi[k] x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest

%o (PARI) a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(gcd(k, n)-1)); \\ _Seiichi Manyama_, Mar 13 2021

%Y Cf. A000010, A055225, A164941, A264782, A279789.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Jul 25 2019