A342470 a(n) = Sum_{d|n} phi(d)^4.

1, 2, 17, 18, 257, 34, 1297, 274, 1313, 514, 10001, 306, 20737, 2594, 4369, 4370, 65537, 2626, 104977, 4626, 22049, 20002, 234257, 4658, 160257, 41474, 106289, 23346, 614657, 8738, 810001, 69906, 170017, 131074, 333329, 23634, 1679617, 209954, 352529, 70418

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

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

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

From Amiram Eldar, Nov 13 2022: (Start)

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

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)

Mathematica

a[n_] := DivisorSum[n, EulerPhi[#]^4 &]; Array[a, 40] (* Amiram Eldar, Mar 13 2021 *)

Prog

(PARI) a(n) = sumdiv(n, d, eulerphi(d)^4);
(PARI) a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^3);
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)^4*x^k/(1-x^k)))

Crossrefs

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

Sequence in context: A198534 A032936 A263329 * A368856 A106849 A064697

Adjacent sequences: A342467 A342468 A342469 * A342471 A342472 A342473

Keyword

nonn,mult

Author

Seiichi Manyama, Mar 13 2021

Status

approved