cp's OEIS Frontend

A007691 Multiply-perfect numbers: n divides sigma(n).

%I A007691 M4182 #148 Jul 18 2025 09:34:10
%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 14182439040,31998395520,43861478400,51001180160,66433720320,137438691328,153003540480,403031236608
%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_, 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 Numbers k such that sigma(k) is divisible by all divisors of k, subsequence of A166070. - _Jaroslav Krizek_, Oct 06 2009
%C A007691 A017666(a(n)) = 1. - _Reinhard Zumkeller_, Apr 06 2012
%C A007691 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
%C A007691 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
%C A007691 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
%C A007691 Numbers such that A054024(n) = 0. - _Michel Marcus_, Nov 16 2013
%C A007691 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
%C A007691 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
%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 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A007691 I. Stewart, L'univers des nombres, "Les nombres multiparfaits", Chapter 15, pp. 82-88, Belin-Pour La Science, Paris 2000.
%D A007691 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 141-148.
%D A007691 David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 135-136.
%H A007691 T. D. Noe, <a href="/A007691/b007691.txt">Table of n, a(n) for n=1..1600</a> (using Flammenkamp's data)
%H A007691 Abiodun E. Adeyemi, <a href="https://arxiv.org/abs/1906.05798">A Study of @-numbers</a>, arXiv:1906.05798 [math.NT], 2019.
%H A007691 Anonymous, <a href="http://www-maths.swan.ac.uk/pgrads/bb/project/node26.html">Multiply Perfect Numbers</a> [broken link]
%H A007691 Eric Bach, Gary Miller, and Jeffrey Shallit, <a href="http://www.cs.cmu.edu/~glmiller/Publications/Papers/BMS86.pdf">Sums of divisors perfect numbers and factoring</a>, SIAM J. Comput. 15:4 (1986), pp. 1143-1154.
%H A007691 Robert D. Carmichael, <a href="http://dx.doi.org/10.1090/S0002-9904-1907-01483-0">A table of multiply perfect numbers</a>, Bull. Amer. Math. Soc. 13 (1907), 383-386.
%H A007691 Farideh Firoozbakht and Maximilian F. Hasler, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Hasler/hasler2.html">Variations on Euclid's formula for Perfect Numbers</a>, JIS 13 (2010) #10.3.1.
%H A007691 Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/mpn.html">The Multiply Perfect Numbers Page</a>
%H A007691 Luis H. Gallardo and Olivier Rahavandrainy, <a href="https://arxiv.org/abs/1908.00106">On (unitary) perfect polynomials over F_2 with only Mersenne primes as odd divisors</a>, arXiv:1908.00106 [math.NT], 2019.
%H A007691 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_6">Perfect, Multiply Perfect, and Sociable Numbers</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.
%H A007691 Florian Luca and John Ferdinands, <a href="http://www.jstor.org/stable/27641939">Problem 11090: Sometimes n divides sigma_k(n)</a>, Amer. Math. Monthly 113:4 (2006), pp. 372-373.
%H A007691 Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">Abundancy : Some Resources </a>
%H A007691 Kaitlin Rafferty and Judy Holdener, <a href="https://www.jstor.org/stable/24345300">On the form of perfect and multiperfect numbers</a>, Pi Mu Epsilon Journal, Vol. 13, No. 5 (Fall 2011), pp. 291-298.
%H A007691 Maxie D. Schmidt, <a href="https://arxiv.org/abs/1705.03488">Exact Formulas for the Generalized Sum-of-Divisors Functions</a>, arXiv:1705.03488 [math.NT], 2017. See p. 11.
%H A007691 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Abundancy.html">Abundancy</a>
%H A007691 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HyperperfectNumber.html">Hyperperfect Number</a>.
%H A007691 <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</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}] (* or *)
%t A007691 Transpose[Select[Table[{n, DivisorSigma[-1, n]}, {n, 100000}], IntegerQ[ #[[2]] ]& ] ][[1]]
%t A007691 (* Third program: *)
%t A007691 Select[Range[10^6], IntegerQ@ DivisorSigma[-1, #] &] (* _Michael De Vlieger_, Mar 19 2021 *)
%o A007691 (PARI) for(n=1,1e6,if(sigma(n)%n==0, print1(n", ")))
%o A007691 (Haskell)
%o A007691 a007691 n = a007691_list !! (n-1)
%o A007691 a007691_list = filter ((== 1) . a017666) [1..]
%o A007691 -- _Reinhard Zumkeller_, Apr 06 2012
%o A007691 (Python)
%o A007691 from sympy import divisor_sigma as sigma
%o A007691 def ok(n): return sigma(n, 1)%n == 0
%o A007691 print([n for n in range(1, 10**4) if ok(n)]) # _Michael S. Branicky_, Jan 06 2021
%Y A007691 Complement is A054027. Cf. A000203, A054030.
%Y A007691 Cf. A000396, A005820, A027687, A046060, A046061, for subsequences of terms with quotient sigma(n)/n = 2..6.
%Y A007691 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.
%Y A007691 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.
%Y A007691 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].
%Y A007691 Positions of ones in A017666, A019294, A094701, A227470, of zeros in A054024, A082901, A173438, A272008, A318996, A326194, A341524. Fixed points of A009194.
%Y A007691 Cf. A069926, A330746 (left inverses, when applied to a(n) give n).
%Y A007691 Cf. A007358, A189000, A327158, A332318/A332319 (for analogs) and A046762, A046763, A046764, A055715, A056006, A081756, A214842, A227302, A227306, A245775, A300906, A325639 (other variants).
%Y A007691 Cf. (other related sequences) A007539, A066135, A066961, A093034, A094467, A134639, A145551, A019278, A194771 [= 2*a(n)], A219545, A229110, A262432, A335830, A336849, A341608.
%K A007691 nonn,nice
%O A007691 1,2
%A A007691 _N. J. A. Sloane_, _Robert G. Wilson v_
%E A007691 More terms from _Jud McCranie_ and then from _David W. Wilson_.
%E A007691 Incorrect comment removed and the crossrefs-section reorganized by _Antti Karttunen_, Mar 20 2021