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A007691
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Multiply-perfect numbers: n divides sigma(n).
(Formerly M4182)
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74
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1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 1476304896, 8589869056, 14182439040, 31998395520, 43861478400, 51001180160, 66433720320
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OFFSET
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1,2
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COMMENTS
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sigma(n)/n is in A054030.
Also numbers such that the sum of the reciprocals of the divisors is an integer. - Harvey P. Dale, Jul 24 2001
Luca's solution of problem 11090, which proves that for k>1 there are an infinite number of n such that n divides sigma_k(n), does not apply to this sequence. However, it is conjectured that this sequence is also infinite. - T. D. Noe, Nov 04 2007
Also numbers n such that A007955(n)/A000203(n) is an integer. [From Ctibor O. Zizka, Jan 12 2009]
Numbers k such that sigma(k) is divisible by all divisors of k, subsequence of A166070. [From Jaroslav Krizek, Oct 06 2009]
A017666(a(n)) = 1. [Reinhard Zumkeller, Apr 06 2012]
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 22.
Florian Luca, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113 (2006), 372-373.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 176.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Stewart, L'univers des nombres, "Les nombres multiparfaits", Chapter 15, pp. 82-88, Belin-Pour La Science, Paris 2000.
D. Wells The Penguin Dictionary of Curious and Interesting Numbers, pp. 135-6, Penguin Books 1987.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1600 (using Flammenkamp's data)
Walter Nissen, Abundancy : Some Resources
Achim Flammenkamp, The Multiply Perfect Numbers Page
Anonymous, Multiply Perfect Numbers
Eric Weisstein's World of Mathematics, Abundancy
Eric Weisstein's World of Mathematics, Hyperperfect Number.
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EXAMPLE
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120 is OK because divisors of 120 are {1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120}, the sum of which is 360=120*3.
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MATHEMATICA
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Do[ If[ Mod[ DivisorSigma[1, n], n ] == 0, Print[n] ], {n, 2, 2*10^11} ]
Transpose[ Select[ Table[ {n, DivisorSigma[ -1, n ]}, {n, 100000} ], IntegerQ[ # [[ 2 ] ] ]& ] ][[ 1 ] ]
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PROG
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(PARI) for(n=1, 1e6, if(sigma(n)%n==0, print1(n", ")))
(Haskell)
a007691 n = a007691_list !! (n-1)
a007691_list = filter ((== 1) . a017666) [1..]
-- Reinhard Zumkeller, Apr 06 2012
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CROSSREFS
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Complement is A054027. Cf. A000203, A054024, A054030, A000396, A005820, A027687, A046060, A046061, A065997, A219545.
Sequence in context: A055715 A026031 A002694 * A065997 A006516 A171476
Adjacent sequences: A007688 A007689 A007690 * A007692 A007693 A007694
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane, Robert G. Wilson v
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EXTENSIONS
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More terms from Jud McCranie and then from David W. Wilson.
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STATUS
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approved
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