A033849 Numbers whose prime factors are 3 and 5.

15, 45, 75, 135, 225, 375, 405, 675, 1125, 1215, 1875, 2025, 3375, 3645, 5625, 6075, 9375, 10125, 10935, 16875, 18225, 28125, 30375, 32805, 46875, 50625, 54675, 84375, 91125, 98415, 140625, 151875, 164025, 234375, 253125, 273375, 295245

Offset: 1

Author

Jeff Burch

Keywords

• nonn

Comments

Numbers k such that phi(k) = (8/15)*k. - Benoit Cloitre, Apr 19 2002

Subsequence of A143202. - Reinhard Zumkeller, Sep 13 2011

Links

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

Crossrefs

Cf. A033845, A033846, A033847, A033848, A033850, A033851, A143201, A143202.

Subsequence of A256617.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a033849 n = a033849_list !! (n-1)
a033849_list = f (singleton (3*5)) where
   f s = m : f (insert (3*m) $ insert (5*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

Mathematica

Sort[Flatten[Table[Table[3^j*5^k, {j, 1, 10}], {k, 1, 10}]]] (* Geoffrey Critzer, Dec 07 2014 *)
Select[Range[300000],FactorInteger[#][[All,1]]=={3,5}&] (* Harvey P. Dale, Oct 19 2022 *)

Python

from sympy import integer_log
def A033849(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 15*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

Formula

From Reinhard Zumkeller, Sep 13 2011: (Start)

A143201(a(n)) = 3.

a(n) = 15*A003593(n). (End)

Sum_{n>=1} 1/a(n) = 1/8. - Amiram Eldar, Dec 22 2020

Extensions

Offset and typo in data fixed by Reinhard Zumkeller, Sep 13 2011