A189393 a(n) = phi(n^4).

1, 8, 54, 128, 500, 432, 2058, 2048, 4374, 4000, 13310, 6912, 26364, 16464, 27000, 32768, 78608, 34992, 123462, 64000, 111132, 106480, 267674, 110592, 312500, 210912, 354294, 263424, 682892, 216000, 893730, 524288, 718740, 628864, 1029000, 559872

Offset: 1

Author

Vincenzo Librandi, Apr 21 2011

Keywords

• nonn
• easy
• mult

Links

• Vincenzo Librandi and T. D. Noe, Table of n, a(n) for n = 1..1000 (terms 1..680 from Vincenzo Librandi)

Crossrefs

Cf. A002618 (phi(n^2)), A053191 (phi(n^3)), A238533 (phi(n^5)), A239442 (phi(n^7)), A239443 (phi(n^9)).

Programs

Magma

[ n^3*EulerPhi(n) : n in [1..100] ]

Mathematica

EulerPhi[Range[100]^4] (* T. D. Noe, Dec 27 2011 *)

PARI

vector(66,n,n^3*eulerphi(n))  /* Joerg Arndt, Apr 22 2011 */

Formula

a(n) = n^3*phi(n).

Dirichlet g.f.: zeta(s - 4) / zeta(s - 3). The n-th term of the Dirichlet inverse is n^3 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega=A001221. - Álvar Ibeas, Nov 24 2017

Sum_{k=1..n} a(k) ~ 6*n^5 / (5*Pi^2). - Vaclav Kotesovec, Feb 02 2019

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^5 - p^4 - p + 1)) = 1.15762316629211803144... - Amiram Eldar, Dec 06 2020