%I A003586
%S A003586 1,2,3,4,6,8,9,12,16,18,24,27,32,36,48,54,64,72,81,96,108,128,144,162,
%T A003586 192,216,243,256,288,324,384,432,486,512,576,648,729,768,864,972,1024,
%U A003586 1152,1296,1458,1536,1728,1944,2048,2187,2304,2592,2916,3072,3456,3888
%N A003586 3-smooth numbers: numbers of the form 2^i*3^j with i, j >= 0.
%C A003586 A061987(n)=a(n+1)-a(n), a(A084791(n))=A084789(n), a(A084791(n)+1)=A084790(n). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 03 2003
%C A003586 Successive numbers k such EulerPhi[6 k] == 2 k. [From Artur Jasinski 
               (grafix(AT)csl.pl), Nov 05 2008]
%C A003586 Where record values greater than 1 occur in A088468: A160519(n)=A088468(a(n)). 
               [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 16 
               2009]
%D A003586 R. Blecksmith, M. McCallum and J. L. Selfridge, 3-smooth representations 
               of integers, Amer. Math. Monthly, 105 (1998), 529-543.
%D A003586 J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des 
               Nombres, Problem 654 pp; 85; 287-8, Ellipses Paris 2004.
%D A003586 D. J. Mintz, 2,3 sequence as a binary mixture, Fib. Quarterly, Vol. 19, 
               No 4, Oct 1981, pp. 351-360.
%D A003586 S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; 
               Chelsea, NY, 1962, p. xxiv.
%D A003586 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.
%H A003586 Franklin T. Adams-Watters, <a href="b003586.txt">Table of n, a(n) for 
               n = 1..501</a>
%H A003586 H. W. Lenstra Jr., <a href="http://www.msri.org/publications/ln/msri/
               1998/mandm/lenstra/1/index.html">Harmonic Numbers</a>
%H A003586 I. Peterson, <a href="http://www.sciencenews.org/sn_arc99/1_23_99/mathland.htm">
               Medieval Harmony</a>
%H A003586 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SmoothNumber.html">Link to a section of The World of Mathematics.</
               a>
%F A003586 An asymptotic formula for a(n) is roughly : a(n)= 1/sqrt(6)*EXP(sqrt(2*ln(2)*ln(3)*n)). 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 20 2001
%F A003586 Union of powers of 2 and 3 with n such that psi(n)=2n, where psi(n)=n*Product_(1+1/
               p) over all prime factors p of n. - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Sep 07 2004
%t A003586 Sort[ Flatten[ Table[ 2^i*3^j, {i, 0, 12}, {j, 0, 8} ] ] ]
%t A003586 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)
%t A003586 aa = {}; Do[If[EulerPhi[6 n] == 2 n, AppendTo[aa, n]], {n, 1, 1000}]; 
               aa [From Artur Jasinski (grafix(AT)csl.pl), Nov 05 2008]
%o A003586 (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",")))
%Y A003586 For p-smooth numbers with other values of p, see A051037, A002473, A051038, 
               A080197, A080681, A080682, A080683.
%Y A003586 a(n) = 2^A022328(n)*3^A022329(n). - N. J. A. Sloane, Mar 19 2009
%Y A003586 Cf. A117221, A105420, A062051, A117222, A105420, A117220, A090184.
%Y A003586 Cf. A131096, A131097.
%Y A003586 A088468, A061987. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               May 16 2009]
%Y A003586 Sequence in context: A053640 A097755 A083854 this_sequence A114334 A018690 
               A018452
%Y A003586 Adjacent sequences: A003583 A003584 A003585 this_sequence A003587 A003588 
               A003589
%K A003586 nonn,easy,nice
%O A003586 1,2
%A A003586 Paul.Zimmermann(AT)loria.fr (Paul Zimmermann)
%E A003586 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
%E A003586 Corrected formula from Lekraj Beedassy - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), 
               Mar 19 2009