A342470 - OEIS

%I #25 Jan 20 2024 14:41:48

%S 1,2,17,18,257,34,1297,274,1313,514,10001,306,20737,2594,4369,4370,

%T 65537,2626,104977,4626,22049,20002,234257,4658,160257,41474,106289,

%U 23346,614657,8738,810001,69906,170017,131074,333329,23634,1679617,209954,352529,70418

%N a(n) = Sum_{d|n} phi(d)^4.

%H Seiichi Manyama, <a href="/A342470/b342470.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^3.

%F G.f.: Sum_{k>=1} phi(k)^4 * x^k/(1 - x^k).

%F From _Amiram Eldar_, Nov 13 2022: (Start)

%F Multiplicative with a(p^e) = 1 + ((p-1)^3*(p^(4*e)-1))/(p^3 + p^2 + p + 1).

%F Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(5)/5) * Product_{p prime} (1 - 4/p^2 + 6/p^3 - 4/p^4 + 1/p^5) = 0.05936545607... . (End)

%t a[n_] := DivisorSum[n, EulerPhi[#]^4 &]; Array[a, 40] (* _Amiram Eldar_, Mar 13 2021 *)

%o (PARI) a(n) = sumdiv(n, d, eulerphi(d)^4);

%o (PARI) a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^3);

%o (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)^4*x^k/(1-x^k)))

%Y Cf. A000010, A013663, A029939, A338997, A342471.

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, Mar 13 2021