A372671 a(n) = phi(6 * n)/2.

1, 2, 3, 4, 4, 6, 6, 8, 9, 8, 10, 12, 12, 12, 12, 16, 16, 18, 18, 16, 18, 20, 22, 24, 20, 24, 27, 24, 28, 24, 30, 32, 30, 32, 24, 36, 36, 36, 36, 32, 40, 36, 42, 40, 36, 44, 46, 48, 42, 40, 48, 48, 52, 54, 40, 48, 54, 56, 58, 48, 60, 60, 54, 64, 48, 60, 66, 64, 66, 48, 70, 72, 72, 72, 60

Offset: 1

Author

Seiichi Manyama, May 10 2024

Keywords

• nonn
• mult
• easy

Comments

The number of integers k from 1 to n such that gcd(n,k) is a 3-smooth number (A003586). - Amiram Eldar, May 18 2025

Crossrefs

Partial sums gives A372637.

Column k=6 of A372673.

Cf. A003586, A008683.

Programs

Mathematica

a[n_] := EulerPhi[6*n]/2; Array[a, 100] (* Amiram Eldar, May 18 2025 *)

PARI

a(n) = eulerphi(6*n)/2;

PARI

my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, moebius(6*k)*x^k/(1-x^k)^2))

Formula

G.f.: Sum_{k>=1} mu(6 * k) * x^k / (1 - x^k)^2, where mu() is the Moebius function (A008683).

Multiplicative with a(p^e) = p^e if p = 2 or 3, and (p-1)*p^(e-1) otherwise.