cp's OEIS Frontend

A025613 Numbers of form 3^i*4^j, with i, j >= 0.

%I A025613 #36 Jul 06 2025 10:00:41
%S A025613 1,3,4,9,12,16,27,36,48,64,81,108,144,192,243,256,324,432,576,729,768,
%T A025613 972,1024,1296,1728,2187,2304,2916,3072,3888,4096,5184,6561,6912,8748,
%U A025613 9216,11664,12288,15552,16384,19683,20736,26244,27648,34992,36864,46656
%N A025613 Numbers of form 3^i*4^j, with i, j >= 0.
%C A025613 Subsequence of 3-smooth numbers, cf. A003586.
%H A025613 Reinhard Zumkeller, <a href="/A025613/b025613.txt">Table of n, a(n) for n = 1..10000</a>
%H A025613 F. Javier de Vega, <a href="https://arxiv.org/abs/2003.13378">An extension of Furstenberg's theorem of the infinitude of primes</a>, arXiv:2003.13378 [math.NT], 2020.
%F A025613 Sum_{n>=1} 1/a(n) = (3*4)/((3-1)*(4-1)) = 2. - _Amiram Eldar_, Sep 24 2020
%F A025613 a(n) ~ exp(sqrt(2*log(3)*log(4)*n)) / sqrt(12). - _Vaclav Kotesovec_, Sep 24 2020
%F A025613 a(n) = 3^A025640(n) * 4^A025645(n). - _R. J. Mathar_, Jul 06 2025
%t A025613 n = 10^5; Flatten[Table[3^i*4^j, {i, 0, Log[3, n]}, {j, 0, Log[4, n/3^i]}]] // Sort (* _Amiram Eldar_, Sep 24 2020 *)
%o A025613 (Haskell)
%o A025613 import Data.Set (singleton, deleteFindMin, insert)
%o A025613 a025613 n = a025613_list !! (n-1)
%o A025613 a025613_list = f $ singleton 1
%o A025613    where f s = m : (f $ insert (3*m) $ insert (4*m) s')
%o A025613              where (m, s') = deleteFindMin s
%o A025613 -- _Reinhard Zumkeller_, Jun 01 2011
%o A025613 (PARI) list(lim)=my(v=List(), N); for(n=0, logint(lim\1,3), N=3^n; while(N<=lim, listput(v, N); N<<=2)); Set(v) \\ _Charles R Greathouse IV_, Sep 10 2015
%Y A025613 Subsequence of A003586.
%K A025613 easy,nonn
%O A025613 1,2
%A A025613 _David W. Wilson_