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A003586 3-smooth numbers: numbers of the form 2^i*3^j with i, j >= 0. 181
1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 648, 729, 768, 864, 972, 1024, 1152, 1296, 1458, 1536, 1728, 1944, 2048, 2187, 2304, 2592, 2916, 3072, 3456, 3888 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A061987(n) = a(n + 1) - a(n), a(A084791(n)) = A084789(n), a(A084791(n) + 1) = A084790(n). - Reinhard Zumkeller, Jun 03 2003

Successive numbers k such that phi(6k) = 2k. - Artur Jasinski, Nov 05 2008

Where record values greater than 1 occur in A088468: A160519(n) = A088468(a(n)). - Reinhard Zumkeller, May 16 2009

Also numbers that are divisible by neither 6k - 1 nor 6k + 1, for all k > 0. - Robert G. Wilson v, Oct 26 2010

Also numbers m such that the rooted tree with Matula-Goebel number m has m antichains. The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. The vertices of a rooted tree can be regarded as a partially ordered set, where u<=v holds for two vertices u and v if and only if u lies on the unique path between v and the root. An antichain is a nonempty set of mutually incomparable vertices. Example: m=4 is in the sequence because the corresponding rooted tree is \/=ARB (R is the root) having 4 antichains (A, R, B, AB). - Emeric Deutsch, Jan 30 2012

A204455(3*a(n)) = 3, and only for these numbers. - Wolfdieter Lang, Feb 04 2012

The number of terms less than or equals to n is the Sum_{i = 0…floor(log(2, n))}, floor(log(3, n/2^i) + 1) or the Sum_{i = 0…floor(log(3, n))}, floor(log(2, n/3^i) + 1) which requires fewer terms to compute. - Robert G. Wilson v, Aug 17 2012

Named 3-friables in French. - Michel Marcus, Jul 17 2013

In the 14th century Levi Ben Gerson proved that the only pairs of terms which differ by 1 are (1,2), (2,3), (3,4), and (8,9); see A235365, A235366, A236210. - Jonathan Sondow, Jan 20 2014

Range of values of A000005(n) (and also A181819(n)) for cubefree numbers n. - Matthew Vandermast, May 14 2014

A036561 is a permutation of this sequence. - L. Edson Jeffery, Sep 22 2014

REFERENCES

J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 654 pp; 85; 287-8, Ellipses Paris 2004.

S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. xxiv.

R. Tijdeman, Some applications of Diophantine approximation, pp. 261-284 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.

LINKS

Franklin T. Adams-Watters and Lei Zhou, Table of n, a(n) for n = 1..10000 (first 501 terms from Franklin T. Adams-Watters)

R. Blecksmith, M. McCallum and J. L. Selfridge, 3-smooth representations of integers, Amer. Math. Monthly, 105 (1998), 529-543.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.

F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.

I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.

A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 252. Book's website

H. W. Lenstra Jr., Harmonic Numbers

D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.

D. J. Mintz, 2,3 sequence as a binary mixture, Fib. Quarterly, Vol. 19, No 4, Oct 1981, pp. 351-360.

I. Peterson, Medieval Harmony

Eric Weisstein's World of Mathematics, Smooth Number

Benoit Cloitre  Plot of a(n)/(1/sqrt(6)*exp(sqrt(2*ln(2)*ln(3)*n)))

FORMULA

An asymptotic formula for a(n) is roughly : a(n) = 1/sqrt(6)*exp(sqrt(2*log(2)*log(3)*n)). - Benoit Cloitre, Nov 20 2001

Union of powers of 2 and 3 with n such that psi(n) = 2*n, where psi(n) = n*Product_(1 + 1/p) over all prime factors p of n. - Lekraj Beedassy, Sep 07 2004

a(n) = 2^A022328(n)*3^A022329(n). - N. J. A. Sloane, Mar 19 2009

The characteristic function of this sequence is given by:

  Sum_{n >= 1} x^a(n) = Sum_{n >= 1} moebius(6*n)*x^n/(1 - x^n). - Paul D. Hanna, Sep 18 2011

MAPLE

A003586 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do numtheory[factorset](a) minus {2, 3} ; if % = {} then return a; end if; end do: end if; end proc: # R. J. Mathar, Feb 28 2011

with(numtheory): for i from 1 to 23328 do if(i/phi(i)=3)then print(i/6) fi od; # Gary Detlefs, Jun 28 2011

MATHEMATICA

a[1] = 1; j = 1; k = 1; n = 100; For[k = 2, k <= n, k++, If[2*a[k - j] < 3^j, a[k] = 2*a[k - j], {a[k] = 3^j, j++}]]; Table[a[i], {i, 1, n}] (* Hai He (hai(AT)mathteach.net) and Gilbert Traub, Dec 28 2004 *)

aa = {}; Do[If[EulerPhi[6 n] == 2 n, AppendTo[aa, n]], {n, 1, 1000}]; aa (* Artur Jasinski, Nov 05 2008 *)

fQ[n_] := Union[ MemberQ[{1, 5}, # ] & /@ Union@ Mod[ Rest@ Divisors@ n, 6]] == {False}; fQ[1] = True; Select[ Range@ 4000, fQ] (* Robert G. Wilson v, Oct 26 2010 *)

powerOfTwo = 12; Select[Nest[Union@Join[#, 2*#, 3*#] &, {1}, powerOfTwo-1], # < 2^powerOfTwo &] (* Robert G. Wilson v and T. D. Noe, Mar 03 2011 *)

fQ[n_] := n == 3 EulerPhi@ n; Select[6 Range@ 4000, fQ]/6 (* Robert G. Wilson v, Jul 08 2011 *)

mx = 4000; Sort@ Flatten@ Table[2^i*3^j, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}] (* Robert G. Wilson v, Aug 17 2012 *)

f[n_] := Block[{p2, p3 = 3^Range[0, Floor@ Log[3, n] + 1]}, p2 = 2^Floor[Log[2, n/p3] + 1]; Min[ Select[ p2*p3, IntegerQ]]]; NestList[f, 1, 54] (* Robert G. Wilson v, Aug 22 2012 *)

PROG

(PARI) test(n)= {m=n; for(p=2, 3, while(m%p==0, m=m/p)); return(m==1)} for(n=1, 4000, if(test(n), print1(n", ")))

(PARI) list(lim)=my(v=List(), N); for(n=0, log(lim\1+.5)\log(3), N=3^n; while(N<=lim, listput(v, N); N<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

(Haskell)

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

smooth :: Set Integer -> [Integer]

smooth s = x : smooth (insert (3*x) $ insert (2*x) s')

  where (x, s') = deleteFindMin s

a003586_list = smooth (singleton 1)

a003586 n = a003586_list !! (n-1)

-- Reinhard Zumkeller, Dec 16 2010

(Sage)

def isA003586(n) :

    return [] == filter(lambda d: d != 2 and d != 3, prime_divisors(n))

@CachedFunction

def A003586(n) :

    if n == 1 : return 1

    k = A003586(n-1) + 1

    while not isA003586(k) : k += 1

    return k

[A003586(n) for n in (1..55)]  # Peter Luschny, Jul 20 2012

(MAGMA) [n: n in [1..4000] | PrimeDivisors(n) subset [2, 3]]; // Bruno Berselli, Sep 24 2012

CROSSREFS

Cf. A051037, A002473, A051038, A080197, A080681, A080682, A117221, A105420, A062051, A117222, A105420, A117220, A090184, A131096, A131097, A186711, A186712, A186771, A088468, A061987, A080683 (p-smooth numbers with other values of p), A025613 (a subsequence).

Cf. also A235365, A235366, A236210.

Cf. A036561

Sequence in context: A053640 A097755 A083854 * A114334 A018690 A018452

Adjacent sequences:  A003583 A003584 A003585 * A003587 A003588 A003589

KEYWORD

nonn,easy,nice

AUTHOR

Paul Zimmermann, Dec 11 1996

EXTENSIONS

Deleted claim that this sequence is union of 2^n (A000079) and 3^n (A000244) sequences - this does not include the terms which are not pure powers. - Walter Roscello (wroscello(AT)comcast.net), Nov 16 2008

Corrected formula from Lekraj Beedassy - Franklin T. Adams-Watters, Mar 19 2009

STATUS

approved

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Last modified December 22 02:02 EST 2014. Contains 252326 sequences.