A065332 3-smooth numbers in their natural position, gaps filled with 0.

1, 2, 3, 4, 0, 6, 0, 8, 9, 0, 0, 12, 0, 0, 0, 16, 0, 18, 0, 0, 0, 0, 0, 24, 0, 0, 27, 0, 0, 0, 0, 32, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 0, 81, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 96, 0, 0

Offset: 1

Author

Reinhard Zumkeller, Oct 29 2001

Keywords

• mult
• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003586, A065330, A065331, A065333, A065334, A065335.

Programs

Mathematica

smooth3Q[n_] := n == 2^IntegerExponent[n, 2]*3^IntegerExponent[n, 3];
a[n_] := n Boole[smooth3Q[n]];
Array[a, 100] (* Jean-François Alcover, Oct 17 2021 *)

PARI

a(n) = if(n >> valuation(n, 2) == 3^valuation(n, 3), n, 0); \\ Amiram Eldar, Sep 16 2023

Formula

a(n) = if A065330(n) = 1 then n else 0.

a(n) = A065333(n) * n.

If a(k) > 0 then a(k) = (2^A065334(k)) * (3^A065335(k)).

From Amiram Eldar, Sep 16 2023: (Start)

Multiplicative with a(p^e) = p^e if p <= 3, and 0 otherwise.

Dirichlet g.f.: 6^s / ((2^s-2)*(3^s-3)).

Sum_{k=1..n} a(k) ~ (n/(log(2)*log(3))) * (log(n) + log(6)/2 - 1). (End)