A216913 a(n) = Gauss_primorial(3*n, 3) / Gauss_primorial(3*n, 3*n).

1, 2, 1, 2, 5, 2, 7, 2, 1, 10, 11, 2, 13, 14, 5, 2, 17, 2, 19, 10, 7, 22, 23, 2, 5, 26, 1, 14, 29, 10, 31, 2, 11, 34, 35, 2, 37, 38, 13, 10, 41, 14, 43, 22, 5, 46, 47, 2, 7, 10, 17, 26, 53, 2, 55, 14, 19, 58, 59, 10, 61, 62, 7, 2, 65, 22, 67, 34, 23, 70, 71

Offset

1,2

Comments

The term Gauss primorial was introduced in A216914 and denotes the restriction of the Gauss factorial N_n! (see A216919) to prime factors.

Multiplicative because both A007947 and A109007 are. - Andrew Howroyd, Aug 02 2018

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Formula

a(n) = n/Sum_{k=1..3n} floor(cos^2(Pi*k^(3n)/(3n))). - Anthony Browne, May 24 2016

a(n) = A007947(n)/A109007(n). - Andrew Howroyd, Aug 02 2018

a(n) = Sum_{d|n} phi(d)*mu(3d)^2. - Ridouane Oudra, Oct 19 2019

From Amiram Eldar, Nov 17 2022: (Start)

Multiplicative with a(3^e) = 1, and a(p^e) = p for p != 3.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (9/22) * Product_{p prime} (1 - 1/(p*(p+1))) = (9/22) * A065463 = 0.2881809... . (End)

Mathematica

Table[n/Sum[Floor[Cos[Pi k^(3 n)/(3 n)]^2], {k, 3 n}], {n, 71}] (* Michael De Vlieger, May 24 2016 *)
a[n_] := Times @@ (First /@ FactorInteger[n])/GCD[n, 3]; Array[a, 100] (* Amiram Eldar, Nov 17 2022 *)

Prog

(PARI) a(n)={factorback(factor(n)[, 1])/gcd(3, n)} \\ Andrew Howroyd, Aug 02 2018
(Sage)
def Gauss_primorial(N, n):
    return mul(j for j in (1..N) if gcd(j, n) == 1 and is_prime(j))
def A216913(n): return Gauss_primorial(3*n, 3)/Gauss_primorial(3*n, 3*n)
[A216913(n) for n in (1..80)]
(Magma) [&+[EulerPhi(d)*MoebiusMu(3*d)^2:d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Oct 19 2019

Crossrefs

Cf. A007947, A065463, A109007, A204455, A216914, A216919.

Sequence in context: A182436 A064192 A284553 * A124218 A025165 A345278

Adjacent sequences: A216910 A216911 A216912 * A216914 A216915 A216916

Keyword

nonn,mult

Author

Peter Luschny, Oct 02 2012

Status

approved