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A003586
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3-smooth numbers: numbers of the form 2^i*3^j with i, j >= 0.
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173
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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
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1,2
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COMMENTS
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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. [From 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 equal 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
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REFERENCES
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R. Blecksmith, M. McCallum and J. L. Selfridge, 3-smooth representations of integers, Amer. Math. Monthly, 105 (1998), 529-543.
J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 654 pp; 85; 287-8, Ellipses Paris 2004.
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.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273
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.
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LINKS
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Franklin T. Adams-Watters and Lei Zhou, Table of n, a(n) for n = 1..10000 (first 501 terms from Franklin T. Adams-Watters)
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
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. 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)))
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FORMULA
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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]
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MAPLE
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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; # From Gary Detlefs, Jun 28 2011
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MATHEMATICA
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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 (* From 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] (* From 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 *)
f[n_] := Block[{p2, p3 = 3^Range[0, Floor@ Log[3, n/2] + 1]}, p2 = Floor[Log[2, n/p3] + 1]; Min[p2*p3]]; NestList[f, 1, 54] (* Lei Zhou, Nov 24 2012 *)
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PROG
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(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
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CROSSREFS
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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).
Sequence in context: A053640 A097755 A083854 * A114334 A018690 A018452
Adjacent sequences: A003583 A003584 A003585 * A003587 A003588 A003589
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Paul.Zimmermann(AT)loria.fr (Paul Zimmermann)
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EXTENSIONS
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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
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STATUS
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approved
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