|
%I #51 Sep 08 2022 08:44:32
%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
%Y Cf. A033849, A112751-A112756, A143202.
%Y Cf. A264997 (partitions into), see also A264998.
%K nonn
%O 1,2
%A _N. J. A. Sloane_
|
