OFFSET
1,1
COMMENTS
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).
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.
Numbers k such that Sum_{d|k} 1/d = 2. - Benoit Cloitre, Apr 07 2002
For number of divisors of a(n) see A061645(n). Number of digits in a(n) is A061193(n). - Lekraj Beedassy, Jun 04 2004
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
The earliest recorded mention of this sequence is in Euclid's Elements, IX 36, about 300 BC. - Artur Jasinski, Jan 25 2006
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
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
If a(n) is even, then 2*a(n) is in A181595. - Vladimir Shevelev, Nov 07 2010
Except for a(1) = 6, all even terms are of the form 30*k - 2 or 45*k + 1. - Arkadiusz Wesolowski, Mar 11 2012
a(4) = A229381(1) = 8128 is the "Simpsons's perfect number". - Jonathan Sondow, Jan 02 2015
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
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
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
If k is perfect and semiprime, then k = 6. - Alexandra Hercilia Pereira Silva, Aug 30 2021
This sequence lists the fixed points of A001065. - Alois P. Heinz, Mar 10 2024
REFERENCES
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 4.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 19.
Stanley J. Bezuszka, Perfect Numbers (Booklet 3, Motivated Math. Project Activities), Boston College Press, Chestnut Hill MA, 1980.
Euclid, Elements, Book IX, Section 36, about 300 BC.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B1.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 239.
T. Koshy, "The Ends Of A Mersenne Prime And An Even Perfect Number", Journal of Recreational Mathematics, Baywood, NY, 1998, pp. 196-202.
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)
József Sándor and Borislav Crstici, Handbook of Number Theory, II, Springer Verlag, 2004.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Ian Stewart, L'univers des nombres, "Diviser Pour Régner", Chapter 14, pp. 74-81, Belin-Pour La Science, Paris, 2000.
Horace S. Uhler, On the 16th and 17th perfect numbers, Scripta Math., Vol. 19 (1953), pp. 128-131.
André Weil, Number Theory, An approach through history, From Hammurapi to Legendre, Birkhäuser, 1984, p. 6.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 107-110, Penguin Books, 1987.
LINKS
E-Hern Lee, Table of n, a(n) for n = 1..15 (terms 1-14 from David Wasserman)
Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.
Anonymous, Perfect Numbers. [broken link]
Anonymous, Timetable of discovery of perfect numbers. [broken link]
Antal Bege and Kinga Fogarasi, Generalized perfect numbers, arXiv:1008.0155 [math.NT], 2010.
Richard P. Brent and Graeme L. Cohen, A new lower bound for odd perfect numbers, Math. Comp., Vol. 53, No. 187 (1989), pp. 431-437, S7; alternative link.
Richard P. Brent, Graeme L. Cohen and Herman J. J. te Riele, A new approach to lower bounds for odd perfect numbers, Report TR-CS-88-08, CSL, ANU, August 1988, 71 pp.
Richard P. Brent, Graeme L. Cohen and Herman J. J. te Riele, Improved Techniques For Lower Bounds For Odd Perfect Numbers, Math. Comp., Vol. 57, No. 196 (1991), pp. 857-868.
J. Britton, Perfect Number Analyser.
C. K. Caldwell, Perfect number.
C. K. Caldwell, Mersenne Primes, etc.
C. K. Caldwell, Iterated sums of the digits of a perfect number converge to 1.
Jose Arnaldo B. Dris, The Abundancy Index of Divisors of Odd Perfect Numbers, J. Int. Seq., Vol. 15 (2012) Article # 12.4.4.
Jason Earls, The Smarandache sum of composites between factors function, in Smarandache Notions Journal, Vol. 14, No. 1 (2004), p. 243.
Roger B. Eggleston, Equisum Partitions of Sets of Positive Integers, Algorithms, Vol. 12, No. 8 (2019), Article 164.
Leonhard Euler, De numeris amicibilibus>, Commentationes arithmeticae collectae, Vol. 2 (1849), pp. 627-636. Written in 1747.
Bakir Farhi, On the representation of an even perfect number as the sum of a limited number of cubes, arXiv:1504.07322 [math.NT], 2015.
Steven Finch, Amicable Pairs and Aliquot Sequences, 2013. [Cached copy, with permission of the author]
Farideh Firoozbakht and Maximilian F. Hasler, Variations on Euclid's formula for Perfect Numbers, Journal of Integer Sequences, Vol. 13 (2010), Article 10.3.1.
J. W. Gaberdiel, A Study of Perfect Numbers and Related Topics.
Takeshi Goto and Yasuo Ohno, Largest prime factor of an odd perfect number, 2006.
Kevin G. Hare, New techniques for bounds on the total number of prime factors of an odd perfect number, Math. Comp., Vol. 76, No. 260 (2007), pp. 2241-2248; arXiv preprint, arXiv:math/0501070 [math.NT], 2005-2006.
Azizul Hoque and Himashree Kalita, Generalized perfect numbers connected with arithmetic functions, Math. Sci. Lett., Vol. 3, No. 3 (2014), pp. 249-253.
C.-E. Jean, "Recreomath" Online Dictionary, Nombre parfait.
Hans-Joachim Kanold, Über die Dichten der Mengen der vollkommenen und der befreundeten Zahlen, Mathematische Zeitschrift, Vol. 61 (1954), pp. 180-185.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, 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.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, The Ramanujan Journal, Vol. 46, No. 3 (2018), pp. 633-655; arXiv preprint, arXiv:1610.07793 [math.NT], 2016.
Pedro Laborde, A Note on the Even Perfect Numbers, The American Mathematical Monthly, Vol. 62, No. 5 (May, 1955), pp. 348-349 (2 pages).
Tom Leinster, Perfect numbers and groups, arXiv:math/0104012 [math.GR], 2001.
A. V. Lelechenko, The Quest for the Generalized Perfect Numbers, in Theoretical and Applied Aspects of Cybernetics, TAAC 2014, Kiev.
Daniel Lustig, The algebraic independence of the sum of divisors functions, Journal of Number Theory, Volume 130, Issue 11 (November 2010), pp. 2628-2633.
T. Masiwa, T. Shonhiwa & G. Hitchcock, Perfect Numbers & Mersenne Primes.
Mathforum, Perfect Numbers.
Mathforum, List of Perfect Numbers.
Judson S. McCranie, A study of hyperperfect numbers, J. Int. Seqs., Vol. 3 (2000), Article #00.1.3.
Gérard P. Michon, Perfect Numbers, Mersenne Primes.
David Moews, Perfect, amicable and sociable numbers.
Derek Muller, The Oldest Unsolved Problem in Math, Veritasium, YouTube video, 2024.
Pace P. Nielsen, Odd perfect numbers have at least nine distinct prime factors, Mathematics of Computation, Vol. 76, No. 260 (2007), pp. 2109-2126; arXiv preprint, arXiv:math/0602485 [math.NT], 2006.
Walter Nissen, Abundancy : Some Resources , 2008-2010.
J. J. O'Connor and E. F. Robertson, Perfect Numbers.
J. O. M. Pedersen, Perfect numbers. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
Ivars Peterson, Cubes of Perfection, MathTrek, 1998.
Paul Pollack, Quasi-Amicable Numbers are Rare, J. Int. Seq., Vol. 14 (2011), Article # 11.5.2.
D. Romagnoli, Perfect Numbers (Text in Italian). [From Lekraj Beedassy, Jun 26 2009]
Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 3.
D. Scheffler and R. Ondrejka, The numerical evaluation of the eighteenth perfect number, Math. Comp., Vol. 14, No. 70 (1960), pp. 199-200.
K. Schneider, perfect number, PlanetMath.org.
Jonathan Sondow and Kieren MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdős-Moser equation, Amer. Math. Monthly, Vol. 124, No. 3 (2017), pp. 232-240; arXiv preprint, arXiv:math/1812.06566 [math.NT], 2018.
G. Villemin's Almanach of Numbers, Nombres Parfaits.
J. Voight, Perfect Numbers:An Elementary Introduction.
Eric Weisstein's World of Mathematics, Perfect Number.
Eric Weisstein's World of Mathematics, Odd Perfect Number.
Eric Weisstein's World of Mathematics, Multiperfect Number.
Eric Weisstein's World of Mathematics, Hyperperfect Number.
Eric Weisstein's World of Mathematics, Abundance.
Wikipedia, Perfect number.
Tomohiro Yamada, On the divisibility of odd perfect numbers by a high power of a prime, arXiv:math/0511410 [math.NT], 2005-2007.
FORMULA
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
a(n) = 2^A133033(n) - 2^A090748(n), assuming there are no odd perfect numbers. - Omar E. Pol, Feb 28 2008
a(n) = A000668(n)*(A000668(n)+1)/2, assuming there are no odd perfect numbers. - Omar E. Pol, Apr 23 2008
a(n) = Sum of the first A000668(n) positive integers, assuming there are no odd perfect numbers. - Omar E. Pol, May 09 2008
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
From Reikku Kulon, Oct 14 2008: (Start)
A144912(2, a(n)) = 1;
A144912(4, a(n)) = -1 for n > 1;
A144912(8, a(n)) = 5 or -5 for all n except 2;
A144912(16, a(n)) = -4 or -13 for n > 1. (End)
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
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
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
a(n) = A275496(2^((A000043(n) - 1)/2)) - 2^A000043(n), assuming there are no odd perfect numbers. - Daniel Poveda Parrilla, Aug 16 2016
a(n) = A156552(A324201(n)), assuming there are no odd perfect numbers. - Antti Karttunen, Mar 28 2019
EXAMPLE
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.
MATHEMATICA
Select[Range[9000], DivisorSigma[1, #]== 2*# &] (* G. C. Greubel, Oct 03 2017 *)
PerfectNumber[Range[15]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 10 2018 *)
PROG
(PARI) isA000396(n) = (sigma(n) == 2*n);
(Haskell)
a000396 n = a000396_list !! (n-1)
a000396_list = [x | x <- [1..], a000203 x == 2 * x]
-- Reinhard Zumkeller, Jan 20 2012
(Python)
from sympy import divisor_sigma
def ok(n): return n > 0 and divisor_sigma(n) == 2*n
print([k for k in range(9999) if ok(k)]) # Michael S. Branicky, Mar 12 2022
CROSSREFS
See A000043 for the current state of knowledge about Mersenne primes.
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.
Cf. A228058 for Euler's criterion for odd terms.
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.
Cf. A001065.
KEYWORD
nonn,nice,core
AUTHOR
EXTENSIONS
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
STATUS
approved