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

1,2

Comments

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

Links

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

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

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 *)

Prog

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

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).

Sequence in context: A057235 A057289 A056754 * A120027 A018586 A135342

Adjacent sequences: A003590 A003591 A003592 * A003594 A003595 A003596

Keyword

nonn

Author

N. J. A. Sloane

Status

approved