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Search: id:A003586
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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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124
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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
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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A061987(n)=a(n+1)-a(n), a(A084791(n))=A084789(n), a(A084791(n)+1)=A084790(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 03 2003
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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.
D. J. Mintz, 2,3 sequence as a binary mixture, Fib. Quarterly, Vol. 19, No 4, Oct 1981, pp. 351-360.
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, Table of n, a(n) for n = 0..500
I. Peterson, Medieval Harmony
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
H. W. Lenstra Jr., Harmonic Numbers
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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*ln(2)*ln(3)*n)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 20 2001
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
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MATHEMATICA
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Sort[ Flatten[ Table[ 2^i*3^j, {i, 0, 12}, {j, 0, 8} ] ] ]
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)
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PROGRAM
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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", ")))
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CROSSREFS
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For p-smooth numbers with other values of p, see A051037, A002473, A051038, A080197, A080681, A080682, A080683.
Cf. A117221, A105420, A062051, A117222, A105420, A117220, A090184.
Cf. A131096, A131097.
Adjacent sequences: A003583 A003584 A003585 this_sequence A003587 A003588 A003589
Sequence in context: A053640 A097755 A083854 this_sequence A114334 A018690 A018452
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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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