%I A005153 M0991
%S A005153 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,
%T A005153 84,88,90,96,100,104,108,112,120,126,128,132,140,144,150,156,160,162,
%U A005153 168,176,180,192,196,198,200,204,208,210,216,220,224,228,234,240,252
%N A005153 Practical numbers (first definition): numbers n such that every k <= 
               sigma(n) is a sum of distinct divisors of n. Also called panarithmic 
               numbers.
%C A005153 Equivalently, numbers n such that every number k <= n is a sum of distinct 
               divisors of n.
%C A005153 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
%C A005153 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
%C A005153 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.
%D A005153 H. Heller, Mathematical Buds, Vol. 1 Chap. 2 pp. 10-22, Mu Alpha Theta 
               OK 1978.
%D A005153 M. R. Heyworth, More on Panarithmic Numbers. New Zealand Math. Mag. 17, 
               28-34 (1980) [ ISSN 0549-0510 ].
%D A005153 H. J. Hindin, Quasipractical numbers, IEEE Communications Magazine, March 
               1980, pp. 41-45.
%D A005153 R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.
%D A005153 E. J. Scourfield, J. Number Theory 62 (1) (1997) p. 163 uses this definition.
%D A005153 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005153 A. K. Srinivasan, Practical numbers, Current Science, 17 (1948), 179-180.
%D A005153 B. M. Stewart, Sums of distinct divisors, Amer. J. Math., 76 (1954), 
               779-785.
%D A005153 See also Math. Rev. 96i:11106.
%H A005153 T. D. Noe, <a href="b005153.txt">Table of n, a(n) for n=1..1000</a>
%H A005153 C. McLeman, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
               PracticalNumber.html">Practical number</a>
%H A005153 G. Melfi, <a href="http://www.dm.unipi.it/gauss-pages/melfi/public_html/
               pratica.html">Practical Numbers</a>
%H A005153 G. Melfi, <a href="http://www.unine.ch/statistics/melfi/pratica.html">
               practical Numbers</a>
%H A005153 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PracticalNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A005153 Wikipedia, <a href="http://en.wikipedia.org/wiki/Practical_number">Practical 
               number</a>
%Y A005153 Cf. A007620 (second definition), A030057.
%Y A005153 Sequence in context: A103288 A125225 A092903 this_sequence A068563 A124240 
               A068997
%Y A005153 Adjacent sequences: A005150 A005151 A005152 this_sequence A005154 A005155 
               A005156
%K A005153 nonn,nice,easy
%O A005153 1,2
%A A005153 N. J. A. Sloane (njas(AT)research.att.com).
%E A005153 More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004