A342488 - OEIS

%I #25 May 09 2021 08:06:25

%S 1,2,5,6,17,14,37,26,53,82,101,74,145,254,385,162,257,398,325,1218,

%T 1697,1102,485,1058,4497,1874,2645,8394,785,19330,901,2306,14497,4354,

%U 112769,17738,1297,6158,37697,270082,1601,316130,1765,105498,1165441,11134,2117,162050,1681381

%N a(n) = Sum_{d|n} phi(d)^(n/d+1).

%H Seiichi Manyama, <a href="/A342488/b342488.txt">Table of n, a(n) for n = 1..8299</a>

%F a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^gcd(k, n).

%F G.f.: Sum_{k>=1} phi(k)^2 * x^k/(1 - phi(k) * x^k).

%F If p is prime, a(p) = 1 + (p-1)^2 = A002522(p-1).

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

%t a[n_] := DivisorSum[n, EulerPhi[#]^(n/#+1) &]; Array[a, 50] (* _Amiram Eldar_, Mar 14 2021 *)

%o (PARI) a(n) = sumdiv(n, d, eulerphi(d)^(n/d+1));

%o (PARI) a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^gcd(k, n));

%o (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)^2*x^k/(1-eulerphi(k)*x^k)))

%Y Cf. A000010, A002522, A164941, A309369, A342485, A342487.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Mar 13 2021