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Search: id:A005153
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| A005153 |
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Practical numbers (first definition): numbers n such that every k <= sigma(n) is a sum of distinct divisors of n. Also called panarithmic numbers.
(Formerly M0991)
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+0
9
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| 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Equivalently, numbers n such that every number k <= n is a sum of distinct divisors of n.
2^r is a member for all r as every number < = sigma(2^r) = 2^(r+1)-1 is a sum of a distinct subset of divisors {1,2,2^2,...2^n}. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 23 2004
Also, numbers n such that A030057(n) > n. This is a consequence of the following lemma, found at the McLeman link: An integer m >= 2 with factorization Product_{i=1}^k p_i^e_i with the p_i in ascending order is practical if and only if p_1 = 2 and, for 1 < i <= k, p_i <= sigma(Product_{j < i} p_j^e_j) + 1. - Franklin T. Adams-Watters, Nov 09 2006
Comment from R. J. Mathar, Nov 27 2006: this definition is used in http://www.dm.unipi.it/gauss-pages/melfi/public_html/articoli/jnt.ps but the same author uses the definition in A007620 in his web page http://members.unine.ch/giuseppe.melfi/pratica.html. See also http://citeseer.ist.psu.edu/285.html and http://arXiv.org/abs/math.NT/0404555.
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REFERENCES
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H. Heller, Mathematical Buds, Vol. 1 Chap. 2 pp. 10-22, Mu Alpha Theta OK 1978.
M. R. Heyworth, More on Panarithmic Numbers. New Zealand Math. Mag. 17, 28-34 (1980) [ ISSN 0549-0510 ].
H. J. Hindin, Quasipractical numbers, IEEE Communications Magazine, March 1980, pp. 41-45.
R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.
E. J. Scourfield, J. Number Theory 62 (1) (1997) p. 163 uses this definition.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. K. Srinivasan, Practical numbers, Current Science, 17 (1948), 179-180.
B. M. Stewart, Sums of distinct divisors, Amer. J. Math., 76 (1954), 779-785.
See also Math. Rev. 96i:11106.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
C. McLeman, PlanetMath.org, Practical number
G. Melfi, Practical Numbers
G. Melfi, practical Numbers
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Wikipedia, Practical number
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CROSSREFS
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Cf. A007620 (second definition), A030057.
Sequence in context: A103288 A125225 A092903 this_sequence A068563 A124240 A068997
Adjacent sequences: A005150 A005151 A005152 this_sequence A005154 A005155 A005156
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004
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