

A003586


3smooth numbers: numbers of the form 2^i*3^j with i, j >= 0.


210



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
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OFFSET

1,2


COMMENTS

These numbers were once called "harmonic numbers", see Lenstra links.  N. J. A. Sloane, Jul 03 2015
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 MatulaGoebel number m has m antichains. The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel 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 3friables 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; 2878, 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. 261284 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.


LINKS

Franklin T. AdamsWatters and Lei Zhou, Table of n, a(n) for n = 1..10000 (first 501 terms from Franklin T. AdamsWatters)
R. Blecksmith, M. McCallum and J. L. Selfridge, 3smooth representations of integers, Amer. Math. Monthly, 105 (1998), 529543.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
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
H. W. Lenstra, Jr., Harmonic Numbers and the ABCconjecture, Abstract of talk, May 30, 2001 [Annotated scanned copy]
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. 351360.
I. Peterson, Medieval Harmony
Raphael Schumacher, The Formula for the Distribution of the 3Smooth Numbers, arXiv:1608.06928 [math.NT], 2016.
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(n1)+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}, powerOfTwo1], # < 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 *)
Select[Range@4000, Last@Map[First, FactorInteger@#] <= 3 &] (* Vincenzo Librandi, Aug 25 2016 *)


PROG

(PARI) test(n)=for(p=2, 3, while(n%p==0, n/=p)); n==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
(PARI) is_A003586(n)=n<5vecmax(factor(n, 5)[, 1])<5 \\ M. F. Hasler, Jan 16 2015
(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 !! (n1)
 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(n1) + 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, A117220, A090184, A131096, A131097, A186711, A186712, A186771, A088468, A061987, A080683 (psmooth numbers with other values of p), A025613 (a subsequence).
Cf. also A235365, A235366, A236210, A036561.
Sequence in context: A097755 A083854 A275199 * A114334 A262609 A018690
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. AdamsWatters, Mar 19 2009


STATUS

approved



