cp's OEIS Frontend

A003593 Numbers of the form 3^i*5^j with i, j >= 0.

%I A003593 #60 Oct 23 2024 00:41:50
%S A003593 1,3,5,9,15,25,27,45,75,81,125,135,225,243,375,405,625,675,729,1125,
%T A003593 1215,1875,2025,2187,3125,3375,3645,5625,6075,6561,9375,10125,10935,
%U A003593 15625,16875,18225,19683,28125,30375,32805,46875,50625,54675,59049
%N A003593 Numbers of the form 3^i*5^j with i, j >= 0.
%C A003593 Odd 5-smooth numbers (A051037). - _Reinhard Zumkeller_, Sep 18 2005
%H A003593 Reinhard Zumkeller, <a href="/A003593/b003593.txt">Table of n, a(n) for n = 1..10000</a>
%F A003593 a(n) ~ 1/sqrt(15)*exp(sqrt(2*log(3)*log(5)*n)) asymptotically. - _Benoit Cloitre_, Jan 22 2002
%F A003593 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 A003593 Sum_{n>=1} 1/a(n) = (3*5)/((3-1)*(5-1)) = 15/8. - _Amiram Eldar_, Sep 22 2020
%p A003593 isA003593 := proc(n)
%p A003593     if n = 1 then
%p A003593         true;
%p A003593     else
%p A003593         return (numtheory[factorset](n) minus {3, 5} = {} );
%p A003593     end if;
%p A003593 end proc:
%p A003593 A003593 := proc(n)
%p A003593     option remember;
%p A003593     if n = 1 then
%p A003593         1;
%p A003593     else
%p A003593         for a from procname(n-1)+1 do
%p A003593             if isA003593(a) then
%p A003593                 return a;
%p A003593             end if;
%p A003593         end do:
%p A003593     end if;
%p A003593 end proc:
%p A003593 seq(A003593(n),n=1..30) ; # _R. J. Mathar_, Aug 04 2016
%t A003593 fQ[n_] := PowerMod[15, n, n] == 0; Select[Range[60000], fQ] (* _Bruno Berselli_, Sep 24 2012 *)
%o A003593 (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 A003593 (PARI) is(n)=n==3^valuation(n,3)*5^valuation(n,5) \\ _Charles R Greathouse IV_, Apr 23 2013
%o A003593 (Haskell)
%o A003593 import Data.Set (singleton, deleteFindMin, insert)
%o A003593 a003593 n = a003593_list !! (n-1)
%o A003593 a003593_list = f (singleton 1) where
%o A003593    f s = m : f (insert (3*m) $ insert (5*m) s') where
%o A003593      (m,s') = deleteFindMin s
%o A003593 -- _Reinhard Zumkeller_, Sep 13 2011
%o A003593 (Magma) [n: n in [1..60000] | PrimeDivisors(n) subset [3,5]]; // _Bruno Berselli_, Sep 24 2012
%o A003593 (GAP) Filtered([1..60000],n->PowerMod(15,n,n)=0); # _Muniru A Asiru_, Mar 19 2019
%o A003593 (Python)
%o A003593 from sympy import integer_log
%o A003593 def A003593(n):
%o A003593     def bisection(f,kmin=0,kmax=1):
%o A003593         while f(kmax) > kmax: kmax <<= 1
%o A003593         while kmax-kmin > 1:
%o A003593             kmid = kmax+kmin>>1
%o A003593             if f(kmid) <= kmid:
%o A003593                 kmax = kmid
%o A003593             else:
%o A003593                 kmin = kmid
%o A003593         return kmax
%o A003593     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 A003593     return bisection(f,n,n) # _Chai Wah Wu_, Oct 22 2024
%Y A003593 Cf. A033849, A112751-A112756, A143202, A022337 (list of j), A022336(list of i).
%Y A003593 Cf. A264997 (partitions into), see also A264998. Cf. A108347 (odd 7-smooth).
%K A003593 nonn
%O A003593 1,2
%A A003593 _N. J. A. Sloane_