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

1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 125, 135, 225, 243, 375, 405, 625, 675, 729, 1125, 1215, 1875, 2025, 2187, 3125, 3375, 3645, 5625, 6075, 6561, 9375, 10125, 10935, 15625, 16875, 18225, 19683, 28125, 30375, 32805, 46875, 50625, 54675, 59049

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Odd 5-smooth numbers (A051037). - Reinhard Zumkeller, Sep 18 2005

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

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

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

Programs

GAP

Filtered([1..60000],n->PowerMod(15,n,n)=0); # Muniru A Asiru, Mar 19 2019

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a003593 n = a003593_list !! (n-1)
a003593_list = f (singleton 1) where
   f s = m : f (insert (3*m) $ insert (5*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

Magma

[n: n in [1..60000] | PrimeDivisors(n) subset [3,5]]; // Bruno Berselli, Sep 24 2012

Maple

isA003593 := proc(n)
    if n = 1 then
        true;
    else
        return (numtheory[factorset](n) minus {3, 5} = {} );
    end if;
end proc:
A003593 := proc(n)
    option remember;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA003593(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A003593(n),n=1..30) ; # R. J. Mathar, Aug 04 2016

Mathematica

fQ[n_] := PowerMod[15, n, n] == 0; Select[Range[60000], fQ] (* Bruno Berselli, Sep 24 2012 *)

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

PARI

is(n)=n==3^valuation(n,3)*5^valuation(n,5) \\ Charles R Greathouse IV, Apr 23 2013

Python

from sympy import integer_log
def A003593(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    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))
    return bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

Formula

a(n) ~ 1/sqrt(15)*exp(sqrt(2*log(3)*log(5)*n)) asymptotically. - Benoit Cloitre, Jan 22 2002

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

Sum_{n>=1} 1/a(n) = (3*5)/((3-1)*(5-1)) = 15/8. - Amiram Eldar, Sep 22 2020