cp's OEIS Frontend

A143207 Numbers with distinct prime factors 2, 3, and 5.

%I A143207 #48 Sep 18 2024 03:51:20
%S A143207 30,60,90,120,150,180,240,270,300,360,450,480,540,600,720,750,810,900,
%T A143207 960,1080,1200,1350,1440,1500,1620,1800,1920,2160,2250,2400,2430,2700,
%U A143207 2880,3000,3240,3600,3750,3840,4050,4320,4500,4800,4860
%N A143207 Numbers with distinct prime factors 2, 3, and 5.
%C A143207 Numbers of the form 2^i * 3^j * 5^k with i, j, k > 0. - _Reinhard Zumkeller_, Sep 13 2011
%C A143207 Integers k such that phi(k)/k = 4/15. - _Artur Jasinski_, Nov 07 2008
%H A143207 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 A143207 A001221(a(n)) = 3; A020639(a(n)) = 2; A006530(a(n)) = 5; A143201(a(n)) = 6.
%F A143207 a(n) = 30*A051037(n); A007947(a(n)) = A010869(n). - _Reinhard Zumkeller_, Sep 13 2011
%F A143207 a(n) ~ sqrt(30) * exp((6*log(2)*log(3)*log(5)*n)^(1/3)). - _Vaclav Kotesovec_, Sep 22 2020
%F A143207 Sum_{n>=1} 1/a(n) = 1/8. - _Amiram Eldar_, Sep 24 2020
%t A143207 a = {}; Do[If[EulerPhi[x]/x == 4/15, AppendTo[a, x]], {x, 1, 11664}]; a (* _Artur Jasinski_, Nov 07 2008 *)
%t A143207 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 A143207 (Haskell)
%o A143207 import Data.Set (singleton, deleteFindMin, insert)
%o A143207 a143207 n = a143207_list !! (n-1)
%o A143207 a143207_list = f (singleton (2*3*5)) where
%o A143207    f s = m : f (insert (2*m) $ insert (3*m) $ insert (5*m) s') where
%o A143207      (m,s') = deleteFindMin s
%o A143207 -- _Reinhard Zumkeller_, Sep 13 2011
%o A143207 (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 A143207 (PARI) is(n) = if(n%30,return(0)); my(f=factor(n,6)[,1]); f[#f]<6 \\ _David A. Corneth_, Sep 22 2020
%o A143207 (Magma) [n: n in [1..5000] | PrimeDivisors(n) eq [2,3,5]]; // _Bruno Berselli_, Sep 14 2015
%o A143207 (Python)
%o A143207 from sympy import integer_log
%o A143207 def A143207(n):
%o A143207     def bisection(f,kmin=0,kmax=1):
%o A143207         while f(kmax) > kmax: kmax <<= 1
%o A143207         while kmax-kmin > 1:
%o A143207             kmid = kmax+kmin>>1
%o A143207             if f(kmid) <= kmid:
%o A143207                 kmax = kmid
%o A143207             else:
%o A143207                 kmin = kmid
%o A143207         return kmax
%o A143207     def f(x):
%o A143207         c = n+x
%o A143207         for i in range(integer_log(x,5)[0]+1):
%o A143207             for j in range(integer_log(m:=x//5**i,3)[0]+1):
%o A143207                 c -= (m//3**j).bit_length()
%o A143207         return c
%o A143207     return bisection(f,n,n)*30 # _Chai Wah Wu_, Sep 16 2024
%Y A143207 Cf. A069819.
%Y A143207 Subsequence of A143204 and of A051037.
%K A143207 nonn,easy
%O A143207 1,1
%A A143207 _Reinhard Zumkeller_, Aug 12 2008
%E A143207 New name from _Charles R Greathouse IV_, Sep 14 2015