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

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

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 4.

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 2d ed. 1966, pp. 11-23.

Stanley J. Bezuszka, Perfect Numbers (Booklet 3, Motivated Math. Project Activities), Boston College Press, Chestnut Hill MA, 1980.

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 136-137.

Harold Davenport, The Higher Arithmetic, Cambridge University Press, 8th ed., 2008, p. 14.

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 263.

Euclid, Elements, Book IX, Section 36, about 300 BC.

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.3 Perfect and Amicable Numbers, pp. 82-83.

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.

Ross Honsberger, Ingenuity in Mathematics, Random House, 1970, pp. 113-116.

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).

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 46-48, 244-245.

Hans Rademacher and Otto Toeplitz, The Enjoyment of Mathematics, Princeton Science Library, 1994. See pp. 129-135.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 83-87.

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.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, chapter 4, pages 127-149.

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.

Leonhard Euler, De numeris amicibilibus>, Commentationes arithmeticae collectae, Vol. 2 (1849), pp. 627-636. Written in 1747.

C.-E. Jean, "Recreomath" Online Dictionary, Nombre parfait.

S. Flora Jeba, Anirban Roy, and Manjil P. Saikia, On k-Facile Perfect Numbers, Algebra and Its Applications (ICAA-2023) Springer Proc. Math. Stat., Vol. 474, 111-121. See p. 111.

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.]

J. O. M. Pedersen, Perfect numbers. [Via Internet Archive Wayback-Machine]

Eric Weisstein's World of Mathematics, Abundance.

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

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

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) = ((2^(A000043(n)))^3 - (2^(A000043(n)) - 1)^3 - 1)/6, assuming there are no odd perfect numbers. - Jules Beauchamp, Jun 06 2025

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.

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 *)

(PARI) isA000396(n) = (sigma(n) == 2*n);

(Haskell)

a000396 n = a000396_list !! (n-1)

a000396_list = [x | x <- [1..], a000203 x == 2 * x]

(Python)

from sympy import divisor_sigma

def ok(n): return n > 0 and divisor_sigma(n) == 2*n

See A000043 for the current state of knowledge about Mersenne primes.

Cf. A228058 for Euler's criterion for odd terms.

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

Reference to Albert H. Beiler's book updated by Harvey P. Dale, Jan 13 2025