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Numbers whose prime factors are 3 and 5.
27

%I #36 Oct 22 2024 15:16:39

%S 15,45,75,135,225,375,405,675,1125,1215,1875,2025,3375,3645,5625,6075,

%T 9375,10125,10935,16875,18225,28125,30375,32805,46875,50625,54675,

%U 84375,91125,98415,140625,151875,164025,234375,253125,273375,295245

%N Numbers whose prime factors are 3 and 5.

%C Numbers k such that phi(k) = (8/15)*k. - _Benoit Cloitre_, Apr 19 2002

%C Subsequence of A143202. - _Reinhard Zumkeller_, Sep 13 2011

%H Reinhard Zumkeller, <a href="/A033849/b033849.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Reinhard Zumkeller_, Sep 13 2011: (Start)

%F A143201(a(n)) = 3.

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

%F Sum_{n>=1} 1/a(n) = 1/8. - _Amiram Eldar_, Dec 22 2020

%t Sort[Flatten[Table[Table[3^j*5^k, {j, 1, 10}], {k, 1, 10}]]] (* _Geoffrey Critzer_, Dec 07 2014 *)

%t Select[Range[300000],FactorInteger[#][[All,1]]=={3,5}&] (* _Harvey P. Dale_, Oct 19 2022 *)

%o (Haskell)

%o import Data.Set (singleton, deleteFindMin, insert)

%o a033849 n = a033849_list !! (n-1)

%o a033849_list = f (singleton (3*5)) where

%o f s = m : f (insert (3*m) $ insert (5*m) s') where

%o (m,s') = deleteFindMin s

%o -- _Reinhard Zumkeller_, Sep 13 2011

%o (Python)

%o from sympy import integer_log

%o def A033849(n):

%o def bisection(f,kmin=0,kmax=1):

%o while f(kmax) > kmax: kmax <<= 1

%o while kmax-kmin > 1:

%o kmid = kmax+kmin>>1

%o if f(kmid) <= kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o return kmax

%o 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 return 15*bisection(f,n,n) # _Chai Wah Wu_, Oct 22 2024

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

%Y Subsequence of A256617.

%K nonn,changed

%O 1,1

%A _Jeff Burch_

%E Offset and typo in data fixed by _Reinhard Zumkeller_, Sep 13 2011