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A005153
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Practical numbers: positive integers 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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89
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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
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OFFSET
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1,2
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COMMENTS
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Equivalently, positive integers 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, Apr 23 2004
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
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
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
Conjecture: The sequence a(n)^(1/n) (n=3,4,...) is strictly decreasing to the limit 1. - Zhi-Wei Sun, Jan 12 2013
Conjecture: For any positive rational number r, there are finitely many pairwise distinct practical numbers q(1)..q(k) such that r = Sum_{j=1..k} 1/q(j). For example, 2 = 1/1 + 1/2 + 1/4 + 1/6 + 1/12 with 1, 2, 4, 6 and 12 all practical, and 10/11 = 1/2 + 1/4 + 1/8 + 1/48 + 1/132 + 1/176 with 2, 4, 8, 48, 132 and 176 all practical. - Zhi-Wei Sun, Sep 12 2015
Analogous with the {1 union primes} (A008578), practical numbers form a complete sequence. This is because it contains all powers of 2 as a subsequence. - Frank M Jackson, Jun 21 2016
Sun's 2015 conjecture on the existence of Egyptian fractions with practical denominators for any positive rational number is true. See the link "Egyptian fractions with practical denominators". - David Eppstein, Nov 20 2016
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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 ].
R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.
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.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
Wayne Dymacek, Letter to N. J. A. Sloane, Jun 15 1978.
D. Eppstein, Egyptian fractions with practical denominators, Nov 20, 2016
H. J. Hindin, Quasipractical numbers, IEEE Communications Magazine, March 1980, pp. 41-45.
Maurice Margenstern, Sur les nombres pratiques, (in French), Groupe d'étude en théorie analytique des nombres, 1 (1984-1985), Exposé No. 21, 13 p.
Maurice Margenstern, Les nombres pratiques: théorie, observations et conjectures, Journal of Number Theory, Volume 37, Issue 1, January 1991, Pages 1-36.
C. McLeman, PlanetMath.org, Practical number
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
G. Melfi, On certain positive integer sequences, arXiv:0404555 [math.NT], 2004.
G. Melfi, A survey of practical numbers (<2008)
G. Melfi, Practical Numbers (old link)
Paul Pollack and Lola Thompson, Practical pretenders, arXiv:1201.3168v1 [math.NT], Jan 16, 2012
Carl Pomerance, Lola Thompson, Andreas Weingartner, On integers n for which X^n-1 has a divisor of every degree, arXiv:1511.03357 [math.NT], 2015.
E. Saias, Entiers à diviseurs denses 1, J. Number Theory 62 (1) (1997) p. 163 uses this definition.
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 [MR64800]
Z.-W. Sun, A conjecture on unit fractions involving primes, preprint, 2015.
Peter Taylor, Table of n, a(n) for n = 1..1000000
Andreas Weingartner, Practical numbers and the distribution of divisors, Q. J. Math. 66 (2015), 743 - 758.
Eric Weisstein's World of Mathematics, Practical Number
Wikipedia, Complete sequence
Wikipedia, Practical number
Robert G. Wilson v, Letter to N. J. A. Sloane, date unknown.
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FORMULA
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Weingartner proves that a(n) ~ kn log n, strengthening an earlier result of Saias. In particular, a(n) = kn log n + O(n log log n). - Charles R Greathouse IV, May 10 2013
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MAPLE
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with(numtheory): with(combinat): P:=proc(n) local a, b, k, j, ok; a:=choose([op(divisors(n))]); b:=[]; for k from 1 to nops(a) do b:=[op(b), add(a[k][j], j=1..nops(a[k]))]; od; for j from 0 to n-1 do ok:=0; for k in b do if k=j then ok:=1; break; fi; od;
if ok=0 then break; fi; od; if ok=1 then n; fi; end:
seq(P(i), i=1..252); # Paolo P. Lava, Jun 13 2017
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a005153 n = a005153_list !! (n-1)
a005153_list = filter (\x -> all (p $ a027750_row x) [1..x]) [1..]
where p _ 0 = True
p [] _ = False
p ds'@(d:ds) m = d <= m && (p ds (m - d) || p ds m)
-- Reinhard Zumkeller, Feb 23 2024, Oct 27 2011
(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
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CROSSREFS
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Cf. A002093, A007620 (second definition), A030057, A033630, A174533.
Cf. A027750.
Sequence in context: A103288 A125225 A092903 * A174973 A238443 A231565
Adjacent sequences: A005150 A005151 A005152 * A005154 A005155 A005156
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KEYWORD
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nonn,nice,easy,changed
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004
Erroneous comment removed by T. D. Noe, Nov 14 2010
Definition changed to exclude n=0 explicitly by M. F. Hasler, Jan 19 2013
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STATUS
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approved
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