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A007691 Multiply-perfect numbers: n divides sigma(n).
(Formerly M4182)
1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 1476304896, 8589869056, 14182439040, 31998395520, 43861478400, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608 (list; graph; refs; listen; history; text; internal format)



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

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

The only terms m > 1 of this sequence that are not in A145551 are m for which sigma(m)/m is not a divisor of m. Conjecture: after 1, A323653 lists all such m (and no other numbers). - Antti Karttunen, Mar 19 2021


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, Penguin Books, 1987, pp. 135-136.


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

Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.

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

Achim Flammenkamp, The Multiply Perfect Numbers Page

Luis H. Gallardo and Olivier Rahavandrainy, On (unitary) perfect polynomials over F_2 with only Mersenne primes as odd divisors, arXiv:1908.00106 [math.NT], 2019.

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

Kaitlin Rafferty and Judy Holdener, On the form of perfect and multiperfect numbers, Pi Mu Epsilon Journal, Vol. 13, No. 5 (Fall 2011), pp. 291-298.

Maxie D. Schmidt, Exact Formulas for the Generalized Sum-of-Divisors Functions, arXiv:1705.03488 [math.NT], 2017. See p. 11.

Eric Weisstein's World of Mathematics, Abundancy

Eric Weisstein's World of Mathematics, Hyperperfect Number.

Index entries for sequences where any odd perfect numbers must occur


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.


Do[If[Mod[DivisorSigma[1, n], n] == 0, Print[n]], {n, 2, 2*10^11}] (* or *)

Transpose[Select[Table[{n, DivisorSigma[-1, n]}, {n, 100000}], IntegerQ[ #[[2]] ]& ] ][[1]]

(* Third program: *)

Select[Range[10^6], IntegerQ@ DivisorSigma[-1, #] &] (* Michael De Vlieger, Mar 19 2021 *)


(PARI) for(n=1, 1e6, if(sigma(n)%n==0, print1(n", ")))


a007691 n = a007691_list !! (n-1)

a007691_list = filter ((== 1) . a017666) [1..]

-- Reinhard Zumkeller, Apr 06 2012


from sympy import divisor_sigma as sigma

def ok(n): return sigma(n, 1)%n == 0

print([n for n in range(1, 10**4) if ok(n)]) # Michael S. Branicky, Jan 06 2021


Complement is A054027. Cf. A000203, A054030.

Cf. A000396, A005820, A027687, A046060, A046061, for subsequences of terms with quotient sigma(n)/n = 2..6.

Other subsequences: A046985, A046986, A046987, A047728, A065997, A066289, (A076231, A076233, A076234), A088844, A088845, A088846, A091443, A114887, A166069, A245782, A260508, A306667, (A325021 U A325022), (A325023 U A325024), (A325025 U A325026), A325637, A323653, A330532, (A330533 U A331724), A336702, A341045.

Subsequence of the following sequences: A011775, A071707, A083865, A089748 (after the initial 1), A102783, A166070, A175200, A225110, A226476, A237719, A245774, A246454, A259307, A263928, A282775, A323652, A336745, A340864. Also conjectured to be a subsequence of A005153, of A307740, and after 1 also of A295078.

Various number-theoretical functions applied to these numbers: A088843 [tau], A098203 [phi], A098204 [gcd(a(n),phi(a(n))], A134665 [2-adic valuation], A307741 [sigma], A308423 [product of divisors], A320024 [the odd part], A134740 [omega], A342658 [bigomega], A342659 [smallest prime not dividing], A342660 [largest prime divisor].

Positions of ones in A017666, A019294, A094701, A227470, of zeros in A054024, A082901, A173438, A272008, A318996, A326194, A341524. Fixed points of A009194.

Cf. A069926, A330746 (left inverses, when applied to a(n) give n).

Cf. A007358, A189000, A327158, A332318/A332319 (for analogs) and A046762, A046763, A046764, A055715, A056006, A081756, A214842, A227302, A227306, A245775, A300906, A325639 (other variants).

Cf. (other related sequences) A007539, A066135, A066961, A093034, A094467, A134639, A145551, A019278, A194771 [= 2*a(n)], A219545, A229110, A262432, A335830, A336849, A341608.

Sequence in context: A026031 A002694 A342924 * A348031 A260508 A334410

Adjacent sequences: A007688 A007689 A007690 * A007692 A007693 A007694




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


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

Incorrect comment removed and the crossrefs-section reorganized by Antti Karttunen, Mar 20 2021



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