A033845 Numbers k of the form 2^i*3^j, where i and j >= 1.

6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 162, 192, 216, 288, 324, 384, 432, 486, 576, 648, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2304, 2592, 2916, 3072, 3456, 3888, 4374, 4608, 5184, 5832, 6144, 6912, 7776, 8748, 9216, 10368, 11664

Offset: 1

Author

Jeff Burch

Keywords

• nonn
• easy

Comments

This sequence is easily confused with A003586, which gives numbers of the form 2^i*3^j with i, j >= 0, and is one-sixth of the present sequence. . Don't simply say "numbers of the form 2^i*3^j", but specify which sequence you mean. - N. J. A. Sloane, May 26 2024

Solutions to phi(n)=n/3 [See J-M. de Koninck & A. Mercier, problème 733].

Numbers n such that Sum_{d prime divisor of n} 1/d = 5/6. - Benoit Cloitre, Apr 13 2002

Also n such that Sum_{d|n} mu(d)^2/d = 2. - Benoit Cloitre, Apr 15 2002

Complement of A006899 with respect to A003586. - Reinhard Zumkeller, Sep 25 2008

In the sieve of Eratosthenes, if one crosses numbers off multiple times, these numbers are crossed off twice, first for 2 and then for 3. - Alonso del Arte, Aug 22 2011

Subsequence of A051037. - Reinhard Zumkeller, Sep 13 2011

Numbers n such that Sum_{d|n} A008683(d)*A000041(d) = 7. - Carl Najafi, Oct 19 2011

Numbers n such that Sum_{d|n} A008683(d)*A000700(d) = 2. - Carl Najafi, Oct 20 2011

Solutions to the equation A001615(x) = 2x. - Enrique Pérez Herrero, Jan 02 2012

So these numbers are called Psi-perfect numbers [see J-M. de Koninck & A. Mercier, problème 654]. - Bernard Schott, Nov 20 2020

References

• J-M. de Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Ellipses, 2004, Problème 733, page 94.
• J-M. de Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Ellipses, 2004, Problème 654, page 85.

Links

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

Crossrefs

Subsequence of A000423, A003586, A051037, A256617.

Cf. A001615, A006899, A086410, A086411, A008683, A143201.

Cf. A191475, A191476.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a033845 n = a033845_list !! (n-1)
a033845_list = f (singleton (2*3)) where
   f s = m : f (insert (2*m) $ insert (3*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

Mathematica

mx = 12000; Sort@ Flatten@ Table[2^i*3^j, {i, Log[2, mx]}, {j, Log[3, mx/2^i]}] (* Robert G. Wilson v, Aug 17 2012 *)

PARI

list(lim)=my(v=List(), N); for(n=0, log(lim\2)\log(3), N=6*3^n; while(N<=lim, listput(v, N); N<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jan 02 2012

Python

from sympy import integer_log
def A033845(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((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    return 6*bisection(f,n,n) # Chai Wah Wu, Sep 15 2024

Python

# faster for initial segment of sequence
import heapq
from itertools import islice
def A033845gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield 6*v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(A033845gen(), 50))) # Michael S. Branicky, Sep 18 2024

Formula

Six times the 3-smooth numbers (A003586). - Ralf Stephan, Apr 16 2004

A086411(a(n)) - A086410(a(n)) = 1. - Reinhard Zumkeller, Sep 25 2008

A143201(a(n)) = 2. - Reinhard Zumkeller, Sep 13 2011

a(n) = 2^A191475(n) * 3^A191476(n). - Zak Seidov, Nov 01 2013

Sum_{n>=1} 1/a(n) = 1/2. - Amiram Eldar, Oct 13 2020

Extensions

Edited by N. J. A. Sloane, Jan 31 2010 and May 26 2024.