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%I M4186 N1744 #438 Mar 10 2024 11:09:44
%S 6,28,496,8128,33550336,8589869056,137438691328,2305843008139952128,
%T 2658455991569831744654692615953842176,
%U 191561942608236107294793378084303638130997321548169216
%N Perfect numbers k: k is equal to the sum of the proper divisors of k.
%C A number k is abundant if sigma(k) > 2k (cf. A005101), perfect if sigma(k) = 2k (this sequence), or deficient if sigma(k) < 2k (cf. A005100), where sigma(k) is the sum of the divisors of k (A000203).
%C The numbers 2^(p-1)*(2^p - 1) are perfect, where p is a prime such that 2^p - 1 is also prime (for the list of p's see A000043). There are no other even perfect numbers and it is believed that there are no odd perfect numbers.
%C Numbers k such that Sum_{d|k} 1/d = 2. - _Benoit Cloitre_, Apr 07 2002
%C For number of divisors of a(n) see A061645(n). Number of digits in a(n) is A061193(n). - _Lekraj Beedassy_, Jun 04 2004
%C All terms other than the first have digital root 1 (since 4^2 == 4 (mod 6), we have, by induction, 4^k == 4 (mod 6), or 2*2^(2*k) = 8 == 2 (mod 6), implying that Mersenne primes M = 2^p - 1, for odd p, are of the form 6*t+1). Thus perfect numbers N, being M-th triangular, have the form (6*t+1)*(3*t+1), whence the property N mod 9 = 1 for all N after the first. - _Lekraj Beedassy_, Aug 21 2004
%C The earliest recorded mention of this sequence is in Euclid's Elements, IX 36, about 300 BC. - _Artur Jasinski_, Jan 25 2006
%C Theorem (Euclid, Euler). An even number m is a perfect number if and only if m = 2^(k-1)*(2^k-1), where 2^k-1 is prime. Euler's idea came from Euclid's Proposition 36 of Book IX (see Weil). It follows that every even perfect number is also a triangular number. - _Mohammad K. Azarian_, Apr 16 2008
%C Triangular numbers (also generalized hexagonal numbers) A000217 whose indices are Mersenne primes A000668, assuming there are no odd perfect numbers. - _Omar E. Pol_, May 09 2008, Sep 15 2013
%C If a(n) is even, then 2*a(n) is in A181595. - _Vladimir Shevelev_, Nov 07 2010
%C Except for a(1) = 6, all even terms are of the form 30*k - 2 or 45*k + 1. - _Arkadiusz Wesolowski_, Mar 11 2012
%C a(4) = A229381(1) = 8128 is the "Simpsons's perfect number". - _Jonathan Sondow_, Jan 02 2015
%C Theorem (_Farideh Firoozbakht_): If m is an integer and both p and p^k-m-1 are prime numbers then x = p^(k-1)*(p^k-m-1) is a solution to the equation sigma(x) = (p*x+m)/(p-1). For example, if we take m=0 and p=2 we get Euclid's result about perfect numbers. - _Farideh Firoozbakht_, Mar 01 2015
%C The cototient of the even perfect numbers is a square; in particular, if 2^p - 1 is a Mersenne prime, cototient(2^(p-1) * (2^p - 1)) = (2^(p-1))^2 (see A152921). So, this sequence is a subsequence of A063752. - _Bernard Schott_, Jan 11 2019
%C Euler's (1747) proof that all the even perfect number are of the form 2^(p-1)*(2^p-1) implies that their asymptotic density is 0. Kanold (1954) proved that the asymptotic density of odd perfect numbers is 0. - _Amiram Eldar_, Feb 13 2021
%C If k is perfect and semiprime, then k = 6. - _Alexandra Hercilia Pereira Silva_, Aug 30 2021
%C This sequence lists the fixed points of A001065. - _Alois P. Heinz_, Mar 10 2024
%D Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 4.
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 19.
%D Stanley J. Bezuszka, Perfect Numbers (Booklet 3, Motivated Math. Project Activities), Boston College Press, Chestnut Hill MA, 1980.
%D Euclid, Elements, Book IX, Section 36, about 300 BC.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 239.
%D T. Koshy, "The Ends Of A Mersenne Prime And An Even Perfect Number", Journal of Recreational Mathematics, Baywood, NY, 1998, pp. 196-202.
%D Joseph S. Madachy, Madachy's Mathematical Recreations, New York: Dover Publications, Inc., 1979, p. 149 (First publ. by Charles Scribner's Sons, New York, 1966, under the title: Mathematics on Vacation)
%D József Sándor and Borislav Crstici, Handbook of Number Theory, II, Springer Verlag, 2004.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D Ian Stewart, L'univers des nombres, "Diviser Pour Régner", Chapter 14, pp. 74-81, Belin-Pour La Science, Paris, 2000.
%D Horace S. Uhler, On the 16th and 17th perfect numbers, Scripta Math., Vol. 19 (1953), pp. 128-131.
%D André Weil, Number Theory, An approach through history, From Hammurapi to Legendre, Birkhäuser, 1984, p. 6.
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 107-110, Penguin Books, 1987.
%H E-Hern Lee, <a href="/A000396/b000396.txt">Table of n, a(n) for n = 1..15</a> (terms 1-14 from David Wasserman)
%H Abiodun E. Adeyemi, <a href="https://arxiv.org/abs/1906.05798">A Study of @-numbers</a>, arXiv:1906.05798 [math.NT], 2019.
%H Anonymous, <a href="http://www-maths.swan.ac.uk/pgrads/bb/project/node3.html">Perfect Numbers</a>. [broken link]
%H Anonymous, <a href="http://www-maths.swan.ac.uk/pgrads/bb/project/node43.html">Timetable of discovery of perfect numbers</a>. [broken link]
%H Antal Bege and Kinga Fogarasi, <a href="http://arxiv.org/abs/1008.0155">Generalized perfect numbers</a>, arXiv:1008.0155 [math.NT], 2010.
%H Richard P. Brent and Graeme L. Cohen, <a href="https://doi.org/10.1090/S0025-5718-1989-0968150-2">A new lower bound for odd perfect numbers</a>, Math. Comp., Vol. 53, No. 187 (1989), pp. 431-437, S7; <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub100.html">alternative link</a>.
%H Richard P. Brent, Graeme L. Cohen and Herman J. J. te Riele, <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub106.html">A new approach to lower bounds for odd perfect numbers</a>, Report TR-CS-88-08, CSL, ANU, August 1988, 71 pp.
%H Richard P. Brent, Graeme L. Cohen and Herman J. J. te Riele, <a href="https://doi.org/10.1090/S0025-5718-1991-1094940-3">Improved Techniques For Lower Bounds For Odd Perfect Numbers</a>, Math. Comp., Vol. 57, No. 196 (1991), pp. 857-868.
%H J. Britton, <a href="http://britton.disted.camosun.bc.ca/perfect/jbperfect.htm">Perfect Number Analyser</a>.
%H C. K. Caldwell, <a href="https://t5k.org/glossary/page.php?sort=PerfectNumber">Perfect number</a>.
%H C. K. Caldwell, <a href="http://www.utm.edu/research/primes/mersenne/index.html">Mersenne Primes, etc.</a>
%H C. K. Caldwell, <a href="http://www.utm.edu/research/primes/notes/proofs/Theorem3.html">Iterated sums of the digits of a perfect number converge to 1</a>.
%H Jose Arnaldo B. Dris, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Dris/dris8.html">The Abundancy Index of Divisors of Odd Perfect Numbers</a>, J. Int. Seq., Vol. 15 (2012) Article # 12.4.4.
%H Jason Earls, <a href="https://pdfs.semanticscholar.org/4559/ac50797ddeda688576630c4d92229440a0a3.pdf#page=243">The Smarandache sum of composites between factors function</a>, in Smarandache Notions Journal, Vol. 14, No. 1 (2004), p. 243.
%H Roger B. Eggleston, <a href="https://doi.org/10.3390/a12080164">Equisum Partitions of Sets of Positive Integers</a>, Algorithms, Vol. 12, No. 8 (2019), Article 164.
%H Leonhard Euler, <a href="https://scholarlycommons.pacific.edu/euler-works/798/">De numeris amicibilibus></a>, Commentationes arithmeticae collectae, Vol. 2 (1849), pp. 627-636. Written in 1747.
%H Bakir Farhi, <a href="http://arxiv.org/abs/1504.07322">On the representation of an even perfect number as the sum of a limited number of cubes</a>, arXiv:1504.07322 [math.NT], 2015.
%H Steven Finch, <a href="/A000396/a000396.pdf">Amicable Pairs and Aliquot Sequences</a>, 2013. [Cached copy, with permission of the author]
%H 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>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.3.1.
%H J. W. Gaberdiel, <a href="http://math.arizona.edu/~ura/001/gaberdiel.jw/">A Study of Perfect Numbers and Related Topics</a>.
%H Takeshi Goto and Yasuo Ohno, <a href="http://www.ma.noda.tus.ac.jp/u/tg/perfect.html">Largest prime factor of an odd perfect number</a>, 2006.
%H Kevin G. Hare, <a href="https://doi.org/10.1090/S0025-5718-07-02033-9">New techniques for bounds on the total number of prime factors of an odd perfect number</a>, Math. Comp., Vol. 76, No. 260 (2007), pp. 2241-2248; <a href="https://arxiv.org/abs/math/0501070">arXiv preprint</a>, arXiv:math/0501070 [math.NT], 2005-2006.
%H Azizul Hoque and Himashree Kalita, <a href="http://www.naturalspublishing.com/files/published/1r9c4i46d2gg27.pdf">Generalized perfect numbers connected with arithmetic functions</a>, Math. Sci. Lett., Vol. 3, No. 3 (2014), pp. 249-253.
%H C.-E. Jean, "Recreomath" Online Dictionary, <a href="http://www.recreomath.qc.ca/dict_parfait_nombre.htm">Nombre parfait</a>.
%H Hans-Joachim Kanold, <a href="https://doi.org/10.1007/BF01181341">Über die Dichten der Mengen der vollkommenen und der befreundeten Zahlen</a>, Mathematische Zeitschrift, Vol. 61 (1954), pp. 180-185.
%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1505.07229v3">The zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv:1505.07229v3 [math.AG], 2015. [A later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication below.]
%H Christian Kassel and Christophe Reutenauer, <a href="https://doi.org/10.1007/s11139-018-0011-1">Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, The Ramanujan Journal, Vol. 46, No. 3 (2018), pp. 633-655; <a href="https://arxiv.org/abs/1610.07793">arXiv preprint</a>, arXiv:1610.07793 [math.NT], 2016.
%H Pedro Laborde, <a href="https://www.jstor.org/stable/2307041">A Note on the Even Perfect Numbers</a>, The American Mathematical Monthly, Vol. 62, No. 5 (May, 1955), pp. 348-349 (2 pages).
%H Tom Leinster, <a href="http://arXiv.org/abs/math.GR/0104012">Perfect numbers and groups</a>, arXiv:math/0104012 [math.GR], 2001.
%H A. V. Lelechenko, <a href="https://web.archive.org/web/20190411020829/https://taac.org.ua/files/a2014/proceedings/UA-2-Andrew%20Lelechenko-440.pdf">The Quest for the Generalized Perfect Numbers</a>, in Theoretical and Applied Aspects of Cybernetics, TAAC 2014, Kiev.
%H Daniel Lustig, <a href="http://dx.doi.org/10.1016/j.jnt.2010.03.022">The algebraic independence of the sum of divisors functions</a>, Journal of Number Theory, Volume 130, Issue 11 (November 2010), pp. 2628-2633.
%H T. Masiwa, T. Shonhiwa & G. Hitchcock, <a href="http://uzweb.uz.ac.zw/science/maths/zimaths/51/perfect.htm">Perfect Numbers & Mersenne Primes</a>.
%H Mathforum, <a href="http://mathforum.org/dr.math/faq/faq.perfect.html">Perfect Numbers</a>.
%H Mathforum, <a href="http://mathforum.org/library/drmath/view/51516.html">List of Perfect Numbers</a>.
%H Judson S. McCranie, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html">A study of hyperperfect numbers</a>, J. Int. Seqs., Vol. 3 (2000), Article #00.1.3.
%H Gérard P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#perfect">Perfect Numbers, Mersenne Primes</a>.
%H David Moews, <a href="http://djm.cc/amicable.html">Perfect, amicable and sociable numbers</a>.
%H Derek Muller, <a href="https://www.youtube.com/watch?v=Zrv1EDIqHkY">The Oldest Unsolved Problem in Math</a>, Veritasium, YouTube video, 2024.
%H Pace P. Nielsen, <a href="https://doi.org/10.1090/S0025-5718-07-01990-4">Odd perfect numbers have at least nine distinct prime factors</a>, Mathematics of Computation, Vol. 76, No. 260 (2007), pp. 2109-2126; <a href="http://arxiv.org/abs/math/0602485">arXiv preprint</a>, arXiv:math/0602485 [math.NT], 2006.
%H Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">Abundancy : Some Resources </a>, 2008-2010.
%H J. J. O'Connor and E. F. Robertson, <a href="http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html">Perfect Numbers</a>.
%H J. O. M. Pedersen, <a href="https://web.archive.org/web/20141006120722/http://amicable.homepage.dk/perfect.htm">Perfect numbers</a>. [Via Internet Archive Wayback-Machine]
%H J. O. M. Pedersen, <a href="http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a>. [Broken link]
%H J. O. M. Pedersen, <a href="http://web.archive.org/web/20140502102524/http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a>. [Via Internet Archive Wayback-Machine]
%H J. O. M. Pedersen, <a href="/A063990/a063990.pdf">Tables of Aliquot Cycles</a>. [Cached copy, pdf file only]
%H Ivars Peterson, <a href="https://web.archive.org/web/20080521092525/http://www.maa.org/mathland/mathtrek_5_18_98.html">Cubes of Perfection</a>, MathTrek, 1998.
%H Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%H Paul Pollack, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pollack/pollack3.html">Quasi-Amicable Numbers are Rare</a>, J. Int. Seq., Vol. 14 (2011), Article # 11.5.2.
%H D. Romagnoli, <a href="http://www.mistretta.eu/PDF/I%20numeri%20perfetti.pdf">Perfect Numbers (Text in Italian)</a>. [From _Lekraj Beedassy_, Jun 26 2009]
%H D. Scheffler and R. Ondrejka, <a href="http://dx.doi.org/10.1090/S0025-5718-1960-0112239-6">The numerical evaluation of the eighteenth perfect number</a>, Math. Comp., Vol. 14, No. 70 (1960), pp. 199-200.
%H K. Schneider, <a href="https://planetmath.org/perfectnumber">perfect number</a>, PlanetMath.org.
%H Jonathan Sondow and Kieren MacMillan, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.124.3.232">Primary pseudoperfect numbers, arithmetic progressions, and the Erdős-Moser equation</a>, Amer. Math. Monthly, Vol. 124, No. 3 (2017), pp. 232-240; <a href="http://arxiv.org/abs/1812.06566">arXiv preprint</a>, arXiv:math/1812.06566 [math.NT], 2018.
%H G. Villemin's Almanach of Numbers, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Decompos/SixNbPf.htm">Nombres Parfaits</a>.
%H J. Voight, <a href="http://magma.maths.usyd.edu.au/~voight/notes/perfelem.pdf">Perfect Numbers:An Elementary Introduction</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectNumber.html">Perfect Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OddPerfectNumber.html">Odd Perfect Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MultiperfectNumber.html">Multiperfect Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HyperperfectNumber.html">Hyperperfect Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Abundance.html">Abundance</a>.
%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Perfect_number">Perfect number</a>.
%H Tomohiro Yamada, <a href="http://arXiv.org/abs/math.NT/0511410">On the divisibility of odd perfect numbers by a high power of a prime</a>, arXiv:math/0511410 [math.NT], 2005-2007.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%F The perfect number N = 2^(p-1)*(2^p - 1) is also multiplicatively p-perfect (i.e., A007955(N) = N^p), since tau(N) = 2*p. - _Lekraj Beedassy_, Sep 21 2004
%F a(n) = 2^A133033(n) - 2^A090748(n), assuming there are no odd perfect numbers. - _Omar E. Pol_, Feb 28 2008
%F a(n) = A000668(n)*(A000668(n)+1)/2, assuming there are no odd perfect numbers. - _Omar E. Pol_, Apr 23 2008
%F a(n) = A000217(A000668(n)), assuming there are no odd perfect numbers. - _Omar E. Pol_, May 09 2008
%F a(n) = Sum of the first A000668(n) positive integers, assuming there are no odd perfect numbers. - _Omar E. Pol_, May 09 2008
%F a(n) = A000384(A019279(n)), assuming there are no odd perfect numbers and no odd superperfect numbers. a(n) = A000384(A061652(n)), assuming there are no odd perfect numbers. - _Omar E. Pol_, Aug 17 2008
%F a(n) = A006516(A000043(n)), assuming there are no odd perfect numbers. - _Omar E. Pol_, Aug 30 2008
%F From _Reikku Kulon_, Oct 14 2008: (Start)
%F A144912(2, a(n)) = 1;
%F A144912(4, a(n)) = -1 for n > 1;
%F A144912(8, a(n)) = 5 or -5 for all n except 2;
%F A144912(16, a(n)) = -4 or -13 for n > 1. (End)
%F a(n) = A019279(n)*A000668(n), assuming there are no odd perfect numbers and odd superperfect numbers. a(n) = A061652(n)*A000668(n), assuming there are no odd perfect numbers. - _Omar E. Pol_, Jan 09 2009
%F a(n) = A007691(A153800(n)), assuming there are no odd perfect numbers. - _Omar E. Pol_, Jan 14 2009
%F Even perfect numbers N = K*A000203(K), where K = A019279(n) = 2^(p-1), A000203(A019279(n)) = A000668(n) = 2^p - 1 = M(p), p = A000043(n). - _Lekraj Beedassy_, May 02 2009
%F a(n) = A060286(A016027(n)), assuming there are no odd perfect numbers. - _Omar E. Pol_, Dec 13 2012
%F For n >= 2, a(n) = Sum_{k=1..A065549(n)} (2*k-1)^3, assuming there are no odd perfect numbers. - _Derek Orr_, Sep 28 2013
%F a(n) = A275496(2^((A000043(n) - 1)/2)) - 2^A000043(n), assuming there are no odd perfect numbers. - _Daniel Poveda Parrilla_, Aug 16 2016
%F a(n) = A156552(A324201(n)), assuming there are no odd perfect numbers. - _Antti Karttunen_, Mar 28 2019
%e 6 is perfect because 6 = 1+2+3, the sum of all divisors of 6 less than 6; 28 is perfect because 28 = 1+2+4+7+14.
%t Select[Range[9000], DivisorSigma[1,#]== 2*# &] (* _G. C. Greubel_, Oct 03 2017 *)
%t PerfectNumber[Range[15]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Dec 10 2018 *)
%o (PARI) isA000396(n) = (sigma(n) == 2*n);
%o (Haskell)
%o a000396 n = a000396_list !! (n-1)
%o a000396_list = [x | x <- [1..], a000203 x == 2 * x]
%o -- _Reinhard Zumkeller_, Jan 20 2012
%o (Python)
%o from sympy import divisor_sigma
%o def ok(n): return n > 0 and divisor_sigma(n) == 2*n
%o print([k for k in range(9999) if ok(k)]) # _Michael S. Branicky_, Mar 12 2022
%Y See A000043 for the current state of knowledge about Mersenne primes.
%Y Cf. A007539, A005820, A027687, A046060, A046061, A000668, A090748, A133033, A000217, A000384, A019279, A061652, A006516, A144912, A153800, A007593, A220290, A028499-A028502, A034916, A065549, A275496, A063752, A156552, A152921, A324201.
%Y Cf. A228058 for Euler's criterion for odd terms.
%Y Positions of 0's in A033879 and in A033880.
%Y Subsequence of following sequences: A005835, A006039, A007691, A023196, A043305, A065997, A083207, A109510, A118372, A216782, A246282, A263837, A294900, A333646, A334410, A335267, A336702, A341622, A342922, A344755, A352739, A357462, and (the even terms), of: A005153, A063752, A174973, A336547, A338520.
%Y Cf. A001065.
%K nonn,nice,core
%O 1,1
%A _N. J. A. Sloane_
%E I removed a large number of comments that assumed there are no odd perfect numbers. There were so many it was getting hard to tell which comments were true and which were conjectures. - _N. J. A. Sloane_, Apr 16 2023
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