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%I M0991
%S 1,2,4,6,8,12,16,18,20,24,28,30,32,36,40,42,48,54,56,60,64,66,72,78,
%T 80,84,88,90,96,100,104,108,112,120,126,128,132,140,144,150,156,160,
%U 162,168,176,180,192,196,198,200,204,208,210,216,220,224,228,234,240,252
%N Practical numbers: positive integers n such that every k <= sigma(n) is a sum of distinct divisors of n. Also called panarithmic numbers.
%C Equivalently, positive integers n such that every number k <= n is a sum of distinct divisors of n.
%C 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 Also, numbers n such that A030057(n) > n. This is a consequence of the following theorem (due to Stewart), 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 Practical numbers first appear in Srinivasan's short paper, which contains terms up to 200. Let n be a practical number. He states that (1) if n>2, n is multiple of 4 or 6; (2) sigma(n) >= 2n-1 (A103288); and (3) 2^t n is practical. He also states that highly composite numbers (A002182), perfect numbers (A000396), and primorial numbers (A002110) are practical. - _T. D. Noe_, Apr 02 2010
%C Strengthening a theorem of Hausman and Shapiro, Pollack shows that every n > 3 for which f(n) >= sqrt{e^{gamma} n log log{n}} is a practical number, where f(n) is the largest integer such that all 0 < m < f(n) can be represented as a sum of distinct divisors of n. (By definition, n is practical if and only if f(n) >= n.) - _Jonathan Vos Post_, Jan 16 2012, corrected by _Charles R Greathouse IV_, May 10 2013
%C Conjecture: The sequence a(n)^(1/n) (n=3,4,...) is strictly decreasing to the limit 1. - _Zhi-Wei Sun_, Jan 12 2013
%D H. Heller, Mathematical Buds, Vol. 1 Chap. 2 pp. 10-22, Mu Alpha Theta OK 1978.
%D M. R. Heyworth, More on Panarithmic Numbers. New Zealand Math. Mag. 17, 28-34 (1980) [ ISSN 0549-0510 ].
%D H. J. Hindin, Quasipractical numbers, IEEE Communications Magazine, March 1980, pp. 41-45.
%D R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.
%D E. J. Scourfield, J. Number Theory 62 (1) (1997) p. 163 uses this definition.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A. K. Srinivasan, Practical numbers, Current Science, 17 (1948), 179-180.
%D B. M. Stewart, Sums of distinct divisors, Amer. J. Math., 76 (1954), 779-785.
%H T. D. Noe, <a href="/A005153/b005153.txt">Table of n, a(n) for n = 1..1000</a>
%H C. McLeman, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/PracticalNumber.html">Practical number</a>
%H G. Melfi, <a href="http://dx.doi.org/10.1006/jnth.1996.0012">On two conjectures about practical numbers</a>, J. Number Theory 56 (1996) 205-210 [<a href="http://www.ams.org/mathscinet-getitem?mr=1370203">MR96i:11106</a>].
%H G. Melfi, <a href="http://arXiv.org/abs/math.NT/0404555">On certain positive integer sequences</a>, arXiv:0404555 [math.NT].
%H G. Melfi, <a href="http://citeseer.ist.psu.edu/285.html">A survey of practical numbers</a> (<2008)
%H G. Melfi, <a href="http://www.dm.unipi.it/gauss-pages/melfi/public_html/pratica.html">Practical Numbers</a> (<a href="http://www.unine.ch/statistics/melfi/pratica.html">old link</a>)
%H Paul Pollack and Lola Thompson, <a href="http://arxiv.org/abs/1201.3168">Practical pretenders</a>, arXiv:1201.3168v1 [math.NT], Jan 16, 2012
%H E. Saias, <a href="http://dx.doi.org/10.1006/jnth.1997.2057">Entiers a diviseurs denses 1</a>, J. Number Theory 62 (1) (1997) p. 163 uses this definition.
%H A. K. Srinivasan, <a href="http://www.ias.ac.in/jarch/currsci/17/179.pdf">Practical numbers</a>, Current Science, 17 (1948), 179-180.
%H B. M. Stewart, <a href="http://www.jstor.org/stable/2372651">Sums of distinct divisors</a>, Amer. J. Math., 76 (1954), 779-785 [<a href="http://www.ams.org/mathscinet-getitem?mr=64800">MR64800</a>]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PracticalNumber.html">Practical Number</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Practical_number">Practical number</a>
%F Sais proves that a(n) ~ n log n. - _Charles R Greathouse IV_, May 10 2013
%t PracticalQ[n_] := Module[{f,p,e,prod=1,ok=True}, If[n<1 || (n>1 && OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p,e} = Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1,prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i,Length[p]}]; ok]]]; Select[Range[200], PracticalQ] (* _T. D. Noe_, Apr 02 2010 *)
%o (Haskell)
%o a005153 n = a005153_list !! (n-1)
%o a005153_list = filter f [1..] where
%o f n = and $ map (p [d | d <- [1..n], mod n d == 0]) [1..n]
%o p _ 0 = True
%o p [] _ = False
%o p (d:ds) m | m < d = False
%o | otherwise = p ds (m - d) || p ds m
%o -- _Reinhard Zumkeller_, Oct 27 2011
%o (PARI) is_A005153(n)=bittest(n,0) && return(n==1); my(P=1); n && !for(i=2,#n=factor(n)~,n[1,i]>1+(P*=sigma(n[1,i-1]^n[2,i-1])) && return) \\ - _M. F. Hasler_, Jan 13 2013
%Y Cf. A007620 (second definition), A030057, A033630, A174533.
%K nonn,nice,easy,changed
%O 1,2
%A _N. J. A. Sloane_.
%E More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004
%E Erroneous comment removed by _T. D. Noe_, Nov 14 2010
%E Definition changed to exclude n=0 explicitly by _M. F. Hasler_, Jan 19 2013
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