A003593 - OEIS

%I #60 Oct 23 2024 00:41:50

%S 1,3,5,9,15,25,27,45,75,81,125,135,225,243,375,405,625,675,729,1125,

%T 1215,1875,2025,2187,3125,3375,3645,5625,6075,6561,9375,10125,10935,

%U 15625,16875,18225,19683,28125,30375,32805,46875,50625,54675,59049

%N Numbers of the form 3^i*5^j with i, j >= 0.

%C Odd 5-smooth numbers (A051037). - _Reinhard Zumkeller_, Sep 18 2005

%H Reinhard Zumkeller, <a href="/A003593/b003593.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) ~ 1/sqrt(15)*exp(sqrt(2*log(3)*log(5)*n)) asymptotically. - _Benoit Cloitre_, Jan 22 2002

%F The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(15*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*5)/((3-1)*(5-1)) = 15/8. - _Amiram Eldar_, Sep 22 2020

%p isA003593 := proc(n)

%p     if n = 1 then

%p         true;

%p     else

%p         return (numtheory[factorset](n) minus {3, 5} = {} );

%p     end if;

%p end proc:

%p A003593 := proc(n)

%p     option remember;

%p     if n = 1 then

%p         1;

%p     else

%p         for a from procname(n-1)+1 do

%p             if isA003593(a) then

%p                 return a;

%p             end if;

%p         end do:

%p     end if;

%p end proc:

%p seq(A003593(n),n=1..30) ; # _R. J. Mathar_, Aug 04 2016

%t fQ[n_] := PowerMod[15, n, n] == 0; Select[Range[60000], fQ] (* _Bruno Berselli_, Sep 24 2012 *)

%o (PARI) list(lim)=my(v=List(),N);for(n=0,log(lim)\log(5),N=5^n;while(N<=lim,listput(v,N);N*=3));vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jun 28 2011

%o (PARI) is(n)=n==3^valuation(n,3)*5^valuation(n,5) \\ _Charles R Greathouse IV_, Apr 23 2013

%o (Haskell)

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

%o a003593 n = a003593_list !! (n-1)

%o a003593_list = f (singleton 1) where

%o    f s = m : f (insert (3*m) $ insert (5*m) s') where

%o      (m,s') = deleteFindMin s

%o -- _Reinhard Zumkeller_, Sep 13 2011

%o (Magma) [n: n in [1..60000] | PrimeDivisors(n) subset [3,5]]; // _Bruno Berselli_, Sep 24 2012

%o (GAP) Filtered([1..60000],n->PowerMod(15,n,n)=0); # _Muniru A Asiru_, Mar 19 2019

%o (Python)

%o from sympy import integer_log

%o def A003593(n):

%o     def bisection(f,kmin=0,kmax=1):

%o         while f(kmax) > kmax: kmax <<= 1

%o         while kmax-kmin > 1:

%o             kmid = kmax+kmin>>1

%o             if f(kmid) <= kmid:

%o                 kmax = kmid

%o             else:

%o                 kmin = kmid

%o         return kmax

%o     def f(x): return n+x-sum(integer_log(x//5**i,3)[0]+1 for i in range(integer_log(x,5)[0]+1))

%o     return bisection(f,n,n) # _Chai Wah Wu_, Oct 22 2024

%Y Cf. A033849, A112751-A112756, A143202, A022337 (list of j), A022336(list of i).

%Y Cf. A264997 (partitions into), see also A264998. Cf. A108347 (odd 7-smooth).

%K nonn,changed

%O 1,2

%A _N. J. A. Sloane_