A112751 -id:A112751 - cp's OEIS frontend

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

N. J. A. Sloane

nonn

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

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

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).

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

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

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

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

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

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

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

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

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

A071520 Number of 5-smooth numbers (A051037) <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 10, 10, 11, 12, 12, 13, 13, 14, 14, 14, 14, 15, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 27, 27, 27, 27, 27, 28, 28

Offset: 1

Benoit Cloitre, Jun 02 2002

A 5-smooth number is a number of the form 2^x*3^y*5^z (x,y,z) >= 0.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008683, A051037, A106598, A112751.

Number of p-smooth numbers <= n: A070939 (p=2), A071521 (p=3), this sequence (p=5), A071604 (p=7), A071523 (p=11), A080684 (p=13), A080685 (p=17), A080686 (p=19).

Accumulate[Table[If[Max[FactorInteger[n][[;;,1]]]<6,1,0],{n,80}]] (* Harvey P. Dale, Aug 04 2024 *)

for(n=1,100,print1(sum(k=1,n,if(sum(i=4,n,if(k%prime(i),0,1)),0,1)),","))

a(n)=-sum(k=1,n,moebius(2*3*5*k)*floor(n/k)) \\ Benoit Cloitre, Jun 14 2007

from sympy import integer_log
def A071520(n):
    c = 0
    for i in range(integer_log(n,5)[0]+1):
        for j in range(integer_log(m:=n//5**i,3)[0]+1):
            c += (m//3**j).bit_length()
    return c # Chai Wah Wu, Sep 16 2024

a(n) = Card{ k | A051037(k) <= n }.
Asymptotically : let a = 1/(6*log(2)*log(3)*log(5)) and b = sqrt(30) then a(n) = a*log(b*n)^3 + O(log(n)).
a(n) = -Sum_{k=1,n} mu(30*k)*floor(n/k). - Benoit Cloitre, Jun 14 2007
a(n) = Sum_{i=0..floor(log_5(n))} Sum_{j=0..floor(log_3(n/5^i))} floor(log_2(2*n/(5^i*3^j))). - Ridouane Oudra, Jul 17 2020

Title corrected by Rainer Rosenthal, Aug 30 2020

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

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A071520 Number of 5-smooth numbers (A051037) <= n.

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