A065334 -id:A065334 - cp's OEIS frontend

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

Reinhard Zumkeller, Oct 29 2001

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

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

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 *)

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

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)

A065335 3-exponents to represent 3-smooth numbers (A065332).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Oct 29 2001

nonn

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

Cf. A006519, A007949, A062153, A065332, A065333, A065334.

a[n_] := If[n/2^IntegerExponent[n, 2]/3^(e = IntegerExponent[n, 3]) == 1, e, 0]; Array[a, 100] (* Amiram Eldar, Feb 21 2021 *)

a(n) = A007949(n) * A065333(n).

a(n) = log_3(n / A006519(n)), where log_3 = A062153. For k > 0 with A065332(k) > 0: A065332(k) = (2^A065334(k)) * (3^a(k)).

cp's OEIS Frontend

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

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