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A007691 Multiply-perfect numbers: n divides sigma(n).
(Formerly M4182)
101
1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 1476304896, 8589869056, 14182439040, 31998395520, 43861478400, 51001180160, 66433720320 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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. - Ctibor O. Zizka, Jan 12 2009

Numbers k such that sigma(k) is divisible by all divisors of k, subsequence of A166070. - Jaroslav Krizek, Oct 06 2009

A017666(a(n)) = 1. - Reinhard Zumkeller, Apr 06 2012

Bach, Miller, & Shallit show that this sequence can be recognized in polynomial time with arbitrarily small error by a probabilistic Turing machine; that is, this sequence is in BPP. - Charles R Greathouse IV, Jun 21 2013

Conjecture: If n is such that 2^n-1 is in A066175 then a(n) is a triangular number. - Ivan N. Ianakiev, Aug 26 2013

Conjecture: Every multiply-perfect number is practical (A005153). I've verified this conjecture for the first 5261 terms with abundancy > 2 using Achim Flammenkamp's data. The even perfect numbers are easily shown to be practical, but every practical number > 1 is even, so a weak form says every even multiply-perfect number is practical. - Jaycob Coleman, Oct 15 2013

Numbers such that A054024(n) = 0. - Michel Marcus, Nov 16 2013

Numbers n such that k(n) = A229110(n) = antisigma(n) mod n = A024816(n) mod n = A000217(n) mod n = (n(n+1)/2) mod n = A142150(n). k(n) = n/2 for even n; k(n) = 0 for odd n (for number 1 and eventually odd multiply-perfect numbers n > 1). - Jaroslav Krizek, May 28 2014

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 22.

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.

LINKS

T. D. Noe, Table of n, a(n) for n=1..1600 (using Flammenkamp's data)

Anonymous, Multiply Perfect Numbers [broken link]

Eric Bach, Gary Miller, and Jeffrey Shallit, Sums of divisors perfect numbers and factoring, SIAM J. Comput. 15:4 (1986), pp. 1143-1154.

R. D. Carmichael, A table of multiply perfect numbers, Bull. Amer. Math. Soc. 13 (1907), 383-386.

F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.

Achim Flammenkamp, The Multiply Perfect Numbers Page

Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.

Walter Nissen, Abundancy : Some Resources

Eric Weisstein's World of Mathematics, Abundancy

Eric Weisstein's World of Mathematics, Hyperperfect Number.

EXAMPLE

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.

MATHEMATICA

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 ] ]

PROG

(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

CROSSREFS

Complement is A054027. Cf. A000203, A054024, A054030, A000396, A005820, A027687, A046060, A046061, A065997, A219545.

Sequence in context: A055715 A026031 A002694 * A260508 A065997 A006516

Adjacent sequences:  A007688 A007689 A007690 * A007692 A007693 A007694

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, Robert G. Wilson v

EXTENSIONS

More terms from Jud McCranie and then from David W. Wilson.

STATUS

approved

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Last modified April 30 08:33 EDT 2017. Contains 285645 sequences.