A318666 a(n) = 2^{the 3-adic valuation of n}.

1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 16, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2

Offset: 1

Author

Antti Karttunen, Sep 03 2018

Keywords

• nonn
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000079, A007949, A046644, A317932.

Programs

Magma

[2^Valuation(n, 3): n in [1..100]]; // Vincenzo Librandi, Mar 19 2020

Mathematica

Table[2^IntegerExponent[n, 3], {n, 100}] (* Vincenzo Librandi, Mar 19 2020 *)

PARI

A318666(n) = 2^valuation(n,3);

PARI

A318666(n) = { my(f = factor(n), m=1); for(i=1, #f~, if(3 == f[i,1], m *= 2^f[i,2])); (m); };

Formula

a(n) = 2^A007949(n).

a(n) = A046644(n)/A317932(n).

Multiplicative with a(3^e) = 2^e, a(p^e) = 1 for any other primes.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Nov 17 2022

Dirichlet g.f.: zeta(s)*(3^s-1)/(3^s-2). - Amiram Eldar, Jan 03 2023

More precise asymptotics: Sum_{k=1..n} a(k) ~ 2*n + zeta(log(2)/log(3)) * n^(log(2)/log(3)) / (2*log(2)). - Vaclav Kotesovec, Jun 25 2024