A053192 a(n) is the cototient of n^3.

0, 4, 9, 32, 25, 144, 49, 256, 243, 600, 121, 1152, 169, 1568, 1575, 2048, 289, 3888, 361, 4800, 3969, 5808, 529, 9216, 3125, 9464, 6561, 12544, 841, 19800, 961, 16384, 14157, 20808, 13475, 31104, 1369, 28880, 22815, 38400, 1681, 52920, 1849, 46464

Offset: 1

Author

Labos Elemer, Mar 02 2000

Keywords

• nonn
• easy

Comments

For n^k, n^k - EulerPhi(n^k) = n^(k-1)*(n-EulerPhi(n)), or cototient(n^k) = n^(k-1)*cototient(n). A similar relation holds for Euler totient function.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000010, A051953, A002618, A053650, A053191, A001248, A229099.

Programs

Magma

[n^3-EulerPhi(n^3): n in [1..44]]; // Vincenzo Librandi, Jul 28 2013

Mathematica

Table[(n^3 - EulerPhi[n^3]), {n, 1, 50}] (* Vincenzo Librandi, Jul 27 2013 *)

PARI

a(n) = n^3 - eulerphi(n^3) \\ Michel Marcus, Jul 26 2013

Formula

a(n) = n^2*Cototient(n) = A051953(n^3) = n^3 - EulerPhi(n^3) = Cototient(n^3).

a(prime(n)) = A051953(prime(n)^3) = A001248(n).

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = 1 - 6/Pi^2 (A229099). - Amiram Eldar, Dec 15 2023