A143207 - OEIS

%I #48 Sep 18 2024 03:51:20

%S 30,60,90,120,150,180,240,270,300,360,450,480,540,600,720,750,810,900,

%T 960,1080,1200,1350,1440,1500,1620,1800,1920,2160,2250,2400,2430,2700,

%U 2880,3000,3240,3600,3750,3840,4050,4320,4500,4800,4860

%N Numbers with distinct prime factors 2, 3, and 5.

%C Numbers of the form 2^i * 3^j * 5^k with i, j, k > 0. - _Reinhard Zumkeller_, Sep 13 2011

%C Integers k such that phi(k)/k = 4/15. - _Artur Jasinski_, Nov 07 2008

%H Vaclav Kotesovec, <a href="/A143207/b143207.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

%F A001221(a(n)) = 3; A020639(a(n)) = 2; A006530(a(n)) = 5; A143201(a(n)) = 6.

%F a(n) = 30*A051037(n); A007947(a(n)) = A010869(n). - _Reinhard Zumkeller_, Sep 13 2011

%F a(n) ~ sqrt(30) * exp((6*log(2)*log(3)*log(5)*n)^(1/3)). - _Vaclav Kotesovec_, Sep 22 2020

%F Sum_{n>=1} 1/a(n) = 1/8. - _Amiram Eldar_, Sep 24 2020

%t a = {}; Do[If[EulerPhi[x]/x == 4/15, AppendTo[a, x]], {x, 1, 11664}]; a (* _Artur Jasinski_, Nov 07 2008 *)

%t n = 10^4; Table[2^i*3^j*5^k, {i, 1, Log[2, n]}, {j, 1, Log[3, n/2^i]}, {k, 1, Log[5, n/(2^i*3^j)]}] // Flatten // Sort (* _Amiram Eldar_, Sep 24 2020 *)

%o (Haskell)

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

%o a143207 n = a143207_list !! (n-1)

%o a143207_list = f (singleton (2*3*5)) where

%o    f s = m : f (insert (2*m) $ insert (3*m) $ insert (5*m) s') where

%o      (m,s') = deleteFindMin s

%o -- _Reinhard Zumkeller_, Sep 13 2011

%o (PARI) list(lim)=my(v=List(),s,t); for(i=1,logint(lim\6,5), t=5^i; for(j=1,logint(lim\t\2,3), s=t*3^j; while((s<<=1)<=lim, listput(v,s)))); Set(v) \\ _Charles R Greathouse IV_, Sep 14 2015

%o (PARI) is(n) = if(n%30,return(0)); my(f=factor(n,6)[,1]); f[#f]<6 \\ _David A. Corneth_, Sep 22 2020

%o (Magma) [n: n in [1..5000] | PrimeDivisors(n) eq [2,3,5]]; // _Bruno Berselli_, Sep 14 2015

%o (Python)

%o from sympy import integer_log

%o def A143207(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):

%o         c = n+x

%o         for i in range(integer_log(x,5)[0]+1):

%o             for j in range(integer_log(m:=x//5**i,3)[0]+1):

%o                 c -= (m//3**j).bit_length()

%o         return c

%o     return bisection(f,n,n)*30 # _Chai Wah Wu_, Sep 16 2024

%Y Cf. A069819.

%Y Subsequence of A143204 and of A051037.

%K nonn,easy

%O 1,1

%A _Reinhard Zumkeller_, Aug 12 2008

%E New name from _Charles R Greathouse IV_, Sep 14 2015