A000423 -id:A000423 - cp's OEIS frontend

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

Jeff Burch

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

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.

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

Subsequence of A000423, A003586, A051037, A256617.
Cf. A001615, A006899, A086410, A086411, A008683, A143201.
Cf. A191475, A191476.

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

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 *)

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

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

# 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

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

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

A058383 Primes of form 1+(2^a)*(3^b), a>0, b>0.

Original entry on oeis.org

7, 13, 19, 37, 73, 97, 109, 163, 193, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 139969, 147457, 209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329, 1179649, 1492993

Offset: 1

Labos Elemer, Dec 20 2000

nonn

Prime numbers n such that cos(2*Pi/n) is an algebraic number of a 3-smooth degree, but not a 2-smooth degree. - Artur Jasinski, Dec 13 2006
From Antonio M. Oller-Marcén, Sep 24 2009: (Start)
In this case gcd(a,b) is a power of 2.
A regular polygon of n sides is constructible by paper folding if and only if n=2^r3^sp_1...p_t with p_i being distinct primes of this kind. (End)
Primes in A005109 but not in A092506. - R. J. Mathar, Sep 28 2012
Conjecture: these are the only solutions >=7 to the equation A000010(x) + A000010(x-1) = floor((4*x-3)/3). - Benoit Cloitre, Mar 02 2018
These are also called Pierpont primes. - Harvey P. Dale, Apr 13 2019

Ray Chandler, Table of n, a(n) for n = 1..8378 (terms < 10^1000, first 1000 terms from T. D. Noe)

Cf. A033845, A000423, A125866, A217035.

N:= 10^10: # to get all terms <= N+1
sort(select(isprime, [seq(seq(1+2^a*3^b, a=1..ilog2(N/3^b)), b=1..floor(log[3](N)))])); # Robert Israel, Mar 02 2018

Do[If[Take[FactorInteger[EulerPhi[2n + 1]][[ -1]],1] == {3} && PrimeQ[2n + 1], Print[2n + 1]], {n, 1, 10000}] (* Artur Jasinski, Dec 13 2006 *)
mx = 1500000; s = Sort@ Flatten@ Table[1 + 2^j*3^k, {j, Log[2, mx]}, {k, Log[3, mx/2^j]}]; Select[s, PrimeQ] (* Robert G. Wilson v, Sep 28 2012 *)
Select[Prime[Range[114000]],FactorInteger[#-1][[All,1]]=={2,3}&] (* Harvey P. Dale, Apr 13 2019 *)

Primes of the form 1 + A033845(n).

A007335 MU-numbers: next term is uniquely the product of 2 earlier terms.

Original entry on oeis.org

2, 3, 6, 12, 18, 24, 48, 54, 96, 162, 192, 216, 384, 486, 768, 864, 1458, 1536, 1944, 3072, 3456, 4374, 6144, 7776, 12288, 13122, 13824, 17496, 24576, 31104, 39366, 49152, 55296, 69984, 98304, 118098, 124416, 157464, 196608, 221184, 279936

Offset: 1

N. J. A. Sloane

All terms are 3-smooth. - Reinhard Zumkeller, Aug 13 2015

Empirically, this sequence corresponds to numbers of the form 2^v * 3^w with v = 1 or w = 1 or v and w both odd (see illustration in Links section). - Rémy Sigrist, Feb 16 2023

Clifford A. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 359.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Shyam Sunder Gupta, Ulam Numbers. In: Exploring the Beauty of Fascinating Numbers. Springer Praxis Books(). Springer, Singapore, (2025).
Rémy Sigrist, Scatterplot of the 2-adic valuation of a(n) vs the 3-adic valuation of a(n) for n = 1..50000
Robert G. Wilson v, Note, n.d.

Subsequence of A000423.

Cf. A054540, A003586, A261255.

a007335 n = a007335_list !! (n-1)
a007335_list = 2 : 3 : f [3, 2] (singleton 6 1) where
   f xs m | v == 1    = y : f (y : xs) (g (map (y *) xs) m')
          | otherwise = f xs m'
          where g [] m = m
                g (z:zs) m = g zs $ insertWith (+) z 1 m
                ((y,v),m') = deleteFindMin m
-- Reinhard Zumkeller, Aug 13 2015

function isMU(u, n, h, i, r)
    ur = u[r]; ui = u[i]
    ur <= ui && return h
    if ur * ui > n
        r -= 1
    elseif ur * ui < n
        i += 1
    else
        h && return false
        h = true; i += 1; r -= 1
    end
    isMU(u, n, h, i, r)
end
function MUList(len)
    u = Array{Int, 1}(undef, len)
    u[1] = 2; u[2] = 3; i = 2; n = 2
    while i < len
        n += 1
        if isMU(u, n, false, 1, i)
            i += 1
            u[i] = n
        end
    end
    return u
end
MUList(41) |> println # Peter Luschny, Apr 07 2019

s={2,3}; Do[n=Select[ Table[s[[j]] s[[k]], {j, Length@s}, {k, j+1, Length@s}] // Flatten // Sort // Split, #[[1]] > s[[-1]] && Length[#] == 1 &][[1,1]]; AppendTo[s, n], {39}]; s  (* Jean-François Alcover, Apr 22 2011 *)
Nest[Append[#, SelectFirst[Union@ Select[Tally@ Map[Times @@ # &, Select[Permutations[#, {2}], #1 < #2 & @@ # &]], Last@ # == 1 &][[All, 1]], Function[k, FreeQ[#, k]]]] &, {2, 3}, 39] (* Michael De Vlieger, Nov 16 2017 *)

a(n) = A003586(A261255(n)). - Reinhard Zumkeller, Aug 13 2015

Conjecture: Sum_{n>=1} 1/a(n) = 181/144. - Amiram Eldar, Jul 31 2022

A060301 For n>3: a(n) is a multiple of three distinct earlier terms.

Original entry on oeis.org

2, 3, 5, 30, 180, 300, 450, 1080, 1800, 2700, 3000, 4500, 6480, 6750, 10800, 16200, 18000, 27000, 30000, 38880, 40500, 45000, 64800, 67500, 97200, 101250, 108000, 162000, 180000, 233280, 243000, 270000, 300000, 388800, 405000, 450000, 583200

Offset: 1

Henry Bottomley, Mar 28 2001

nonn

			a(15) = 10800 = 2*30*180 = a(1)*a(4)*a(5).

R. G. Wilson, V, Note, n.d.

Cf. A000423.

a(n) is of the form 2^b*3^c*5^d where either b+c+d=1, or where each of b, c and d are positive and the largest of the three is an odd number less than the sum of the other two. For n>3: a(n) is 30 times a power of 6 times a power of 10 times a power of 15.

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

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A058383 Primes of form 1+(2^a)*(3^b), a>0, b>0.

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A007335 MU-numbers: next term is uniquely the product of 2 earlier terms.

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A060301 For n>3: a(n) is a multiple of three distinct earlier terms.

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