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

%I A065332 #23 Aug 18 2024 18:23:50
%S A065332 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,
%T A065332 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,
%U A065332 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
%N A065332 3-smooth numbers in their natural position, gaps filled with 0.
%H A065332 Amiram Eldar, <a href="/A065332/b065332.txt">Table of n, a(n) for n = 1..10000</a>
%F A065332 a(n) = if A065330(n) = 1 then n else 0.
%F A065332 a(n) = A065333(n) * n.
%F A065332 If a(k) > 0 then a(k) = (2^A065334(k)) * (3^A065335(k)).
%F A065332 From _Amiram Eldar_, Sep 16 2023: (Start)
%F A065332 Multiplicative with a(p^e) = p^e if p <= 3, and 0 otherwise.
%F A065332 Dirichlet g.f.: 6^s / ((2^s-2)*(3^s-3)).
%F A065332 Sum_{k=1..n} a(k) ~ (n/(log(2)*log(3))) * (log(n) + log(6)/2 - 1). (End)
%t A065332 smooth3Q[n_] := n == 2^IntegerExponent[n, 2]*3^IntegerExponent[n, 3];
%t A065332 a[n_] := n Boole[smooth3Q[n]];
%t A065332 Array[a, 100] (* _Jean-François Alcover_, Oct 17 2021 *)
%o A065332 (PARI) a(n) = if(n >> valuation(n, 2) == 3^valuation(n, 3), n, 0); \\ _Amiram Eldar_, Sep 16 2023
%Y A065332 Cf. A003586, A065330, A065331, A065333, A065334, A065335.
%K A065332 mult,nonn,easy
%O A065332 1,2
%A A065332 _Reinhard Zumkeller_, Oct 29 2001