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

1, 3, 4, 9, 12, 16, 27, 36, 48, 64, 81, 108, 144, 192, 243, 256, 324, 432, 576, 729, 768, 972, 1024, 1296, 1728, 2187, 2304, 2916, 3072, 3888, 4096, 5184, 6561, 6912, 8748, 9216, 11664, 12288, 15552, 16384, 19683, 20736, 26244, 27648, 34992, 36864, 46656

Offset: 1

Author

David W. Wilson

Keywords

• easy
• nonn

Comments

Subsequence of 3-smooth numbers, cf. A003586.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.

Crossrefs

Subsequence of A003586.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a025613 n = a025613_list !! (n-1)
a025613_list = f $ singleton 1
   where f s = m : (f $ insert (3*m) $ insert (4*m) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

Mathematica

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 *)

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

Formula

Sum_{n>=1} 1/a(n) = (3*4)/((3-1)*(4-1)) = 2. - Amiram Eldar, Sep 24 2020

a(n) ~ exp(sqrt(2*log(3)*log(4)*n)) / sqrt(12). - Vaclav Kotesovec, Sep 24 2020

a(n) = 3^A025640(n) * 4^A025645(n). - R. J. Mathar, Jul 06 2025