Search: id:A007691
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%I A007691 M4182
%S A007691 1,6,28,120,496,672,8128,30240,32760,523776,2178540,23569920,33550336,
%T A007691 45532800,142990848,459818240,1379454720,1476304896,8589869056,
%U A007691 14182439040,31998395520,43861478400,51001180160,66433720320
%N A007691 Multiply-perfect numbers: n divides sigma(n).
%C A007691 sigma(n)/n is in A054030.
%C A007691 Also numbers such that the sum of the reciprocals of the divisors is 
               an integer. - Harvey P. Dale (hpd1(AT)nyu.edu), Jul 24 2001
%C A007691 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
%C A007691 Also numbers n such that A007955(n)/A000203(n) is an integer. [From Ctibor 
               O. Zizka (c.zizka(AT)email.cz), Jan 12 2009]
%C A007691 Numbers k such that sigma(k) is divisible by all divisors of k, subsequence 
               of A166070. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 
               Oct 06 2009]
%D A007691 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007691 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 22.
%D A007691 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 176.
%D A007691 D. Wells The Penguin Dictionary of Curious and Interesting Numbers, pp. 
               135-6, Penguin Books 1987.
%D A007691 I. Stewart, L'univers des nombres, "Les nombres multiparfaits", Chapter 
               15, pp. 82-88, Belin-Pour La Science, Paris 2000.
%D A007691 Florian Luca, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. 
               Monthly 113 (2006), 372-373.
%H A007691 T. D. Noe, <a href="b007691.txt">Table of n, a(n) for n=1..1600</a> (using 
               Flammenkamp's data)
%H A007691 Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">
               Abundancy : Some Resources </a>
%H A007691 Achim Flammenkamp, <a href="http://www.uni-bielefeld.de/~achim/mpn.html">
               The Multiply Perfect Numbers Page</a>
%H A007691 Anonymous, <a href="http://www-maths.swan.ac.uk/pgrads/bb/project/node26.html">
               Multiply Perfect Numbers</a>
%H A007691 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Abundancy.html">Abundancy</a>
%H A007691 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HyperperfectNumber.html">Hyperperfect Number</a>.
%e A007691 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.
%t A007691 Do[ If[ Mod[ DivisorSigma[1, n], n ] == 0, Print[n] ], {n, 2, 2*10^11} 
               ]
%t A007691 Transpose[ Select[ Table[ {n, DivisorSigma[ -1, n ]}, {n, 100000} ], 
               IntegerQ[ # [[ 2 ] ] ]& ] ][[ 1 ] ]
%Y A007691 Complement is A054027. Cf. A000203, A054024, A054030.
%Y A007691 Cf. A000396, A005820, A027687, A046060, A046061
%Y A007691 Sequence in context: A055715 A026031 A002694 this_sequence A065997 A006516 
               A037131
%Y A007691 Adjacent sequences: A007688 A007689 A007690 this_sequence A007692 A007693 
               A007694
%K A007691 nonn,nice
%O A007691 1,2
%A A007691 N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)
%E A007691 More terms from Jud Mccranie (j.mccranie(AT)comcast.net) and then from 
               David W. Wilson (davidwwilson(AT)comcast.net).

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