%I #34 Oct 22 2023 16:25:36

%S 1,3,9,11,27,33,81,99,121,243,297,363,729,891,1089,1331,2187,2673,

%T 3267,3993,6561,8019,9801,11979,14641,19683,24057,29403,35937,43923,

%U 59049,72171,88209,107811,131769,161051,177147,216513,264627,323433

%N Numbers of the form 3^i*11^j.

%H Reinhard Zumkeller, <a href="/A003597/b003597.txt">Table of n, a(n) for n = 1..10000</a>

%F The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(33*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - _Peter Bala_, Mar 18 2019

%F Sum_{n>=1} 1/a(n) = (3*11)/((3-1)*(11-1)) = 33/20. - _Amiram Eldar_, Sep 23 2020

%F a(n) ~ exp(sqrt(2*log(3)*log(11)*n)) / sqrt(33). - _Vaclav Kotesovec_, Sep 23 2020

%t fQ[n_]:=PowerMod[33, n, n] == 0; Select[Range[4*10^5], fQ] (* _Vincenzo Librandi_, Jun 27 2016 *)

%o (PARI) list(lim)=my(v=List(),N);for(n=0,log(lim)\log(11),N=11^n;while(N<=lim,listput(v,N);N*=3));vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jun 28 2011

%o (Haskell)

%o import Data.Set (singleton, deleteFindMin, insert)

%o a003597 n = a003597_list !! (n-1)

%o a003597_list = f $ singleton (1,0,0) where

%o f s = y : f (insert (3 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')

%o where ((y, i, j), s') = deleteFindMin s

%o -- _Reinhard Zumkeller_, May 15 2015

%o (Magma) [n: n in [1..4*10^5] | PrimeDivisors(n) subset [3, 11]]; // _Vincenzo Librandi_, Jun 27 2016

%o (GAP) Filtered([1..324000],n->PowerMod(33,n,n)=0); # _Muniru A Asiru_, Mar 19 2019

%Y Cf. A025612, A025616, A025621, A025625, A025629, A025632, A025634, A025635, A108761, A003596, A107988, A003598, A108698, A003599, A107788, A108687, A108779, A108090.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_