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

30, 60, 90, 120, 150, 180, 240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810, 900, 960, 1080, 1200, 1350, 1440, 1500, 1620, 1800, 1920, 2160, 2250, 2400, 2430, 2700, 2880, 3000, 3240, 3600, 3750, 3840, 4050, 4320, 4500, 4800, 4860

Offset

1,1

Comments

Numbers of the form 2^i * 3^j * 5^k with i, j, k > 0. - Reinhard Zumkeller, Sep 13 2011

Integers k such that phi(k)/k = 4/15. - Artur Jasinski, Nov 07 2008

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Formula

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

a(n) = 30*A051037(n); A007947(a(n)) = A010869(n). - Reinhard Zumkeller, Sep 13 2011

a(n) ~ sqrt(30) * exp((6*log(2)*log(3)*log(5)*n)^(1/3)). - Vaclav Kotesovec, Sep 22 2020

Sum_{n>=1} 1/a(n) = 1/8. - Amiram Eldar, Sep 24 2020

Mathematica

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

Prog

(Haskell)
import Data.Set (singleton, deleteFindMin, insert)
a143207 n = a143207_list !! (n-1)
a143207_list = f (singleton (2*3*5)) where
   f s = m : f (insert (2*m) $ insert (3*m) $ insert (5*m) s') where
     (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011
(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
(PARI) is(n) = if(n%30, return(0)); my(f=factor(n, 6)[, 1]); f[#f]<6 \\ David A. Corneth, Sep 22 2020
(Magma) [n: n in [1..5000] | PrimeDivisors(n) eq [2, 3, 5]]; // Bruno Berselli, Sep 14 2015
(Python)
from sympy import integer_log
def A143207(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):
        c = n+x
        for i in range(integer_log(x, 5)[0]+1):
            for j in range(integer_log(m:=x//5**i, 3)[0]+1):
                c -= (m//3**j).bit_length()
        return c
    return bisection(f, n, n)*30 # Chai Wah Wu, Sep 16 2024

Crossrefs

Cf. A069819.

Subsequence of A143204 and of A051037.

Sequence in context: A050519 A358756 A069819 * A359410 A108454 A235483

Adjacent sequences: A143204 A143205 A143206 * A143208 A143209 A143210

Keyword

nonn,easy

Author

Reinhard Zumkeller, Aug 12 2008

Extensions

New name from Charles R Greathouse IV, Sep 14 2015

Status

approved