A065997 -id:A065997 - cp's OEIS frontend

A000396 Perfect numbers k: k is equal to the sum of the proper divisors of k.

6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216

Offset: 1

N. J. A. Sloane

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

			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.

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.
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.
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.
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.

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.
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
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.
Chris K. Caldwell, Perfect number.
Chris K. Caldwell, Mersenne Primes, etc.
Chris 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.
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.
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.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
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.
Joshua Zelinsky, The Sum of the Reciprocals of the Prime Divisors of an Odd Perfect or Odd Primitive Non-deficient Number, Integers (2025) Vol. 25, Art. No. A59. See p. 1.
Index entries for "core" sequences
Index entries for sequences where any odd perfect numbers must occur

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.
Positions of 0's in A033879 and in A033880.
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.

a000396 n = a000396_list !! (n-1)
a000396_list = [x | x <- [1..], a000203 x == 2 * x]
-- Reinhard Zumkeller, Jan 20 2012

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);
```

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

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) = A000217(A000668(n)), assuming there are no odd perfect numbers. - Omar E. Pol, May 09 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
a(n) = A006516(A000043(n)), assuming there are no odd perfect numbers. - Omar E. Pol, Aug 30 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
a(n) = A007691(A153800(n)), assuming there are no odd perfect numbers. - Omar E. Pol, Jan 14 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
a(n) = A060286(A016027(n)), assuming there are no odd perfect numbers. - Omar E. Pol, Dec 13 2012
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
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

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

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

Original entry on oeis.org

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

Offset: 1

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

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

			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.

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.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 141-148.
David 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.
Robert D. Carmichael, A table of multiply perfect numbers, Bull. Amer. Math. Soc. 13 (1907), 383-386.
Farideh Firoozbakht and Maximilian 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.
Shyam Sunder Gupta, Perfect, Multiply Perfect, and Sociable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.
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

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.

a007691 n = a007691_list !! (n-1)
a007691_list = filter ((== 1) . a017666) [1..]
-- Reinhard Zumkeller, Apr 06 2012

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 *)

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

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

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

A054030 Sigma(n)/n for n such that sigma(n) is divisible by n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 4, 3, 4, 4, 2, 4, 4, 3, 4, 3, 2, 5, 5, 4, 3, 4, 2, 4, 4, 5, 4, 5, 5, 4, 5, 5, 4, 4, 4, 5, 4, 4, 2, 5, 4, 5, 6, 5, 5, 5, 5, 5, 5, 6, 5, 5, 4, 5, 6, 5, 4, 4, 5, 4, 5, 4, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 5, 6, 5, 6, 6, 5, 4, 4, 5, 4, 4, 5, 6, 5, 5, 4, 6, 4, 4, 6, 5, 6, 6, 6, 6, 6, 6, 6, 5, 6

Offset: 1

Asher Auel, Jan 19 2000

The graph supports the conjecture that all numbers except 2 appear only a finite number of times. Sequences A000396, A005820, A027687, A046060 and A046061 give the n for which the abundancy sigma(n)/n is 2, 3, 4, 5 and 6, respectively. See A134639 for the number of n having abundancy greater than 2. - T. D. Noe, Nov 04 2007

T. D. Noe, Table of n, a(n) for n = 1..1600 (using Flammenkamp's data)
Achim Flammenkamp, The Multiply Perfect Numbers Page
Eric Weisstein's World of Mathematics, Abundancy

Cf. A000203, A007691, A054024, A065997, A219545.

with(numtheory): for i while i < 33000 do
if sigma(i) mod i = 0 then print(sigma(i)/i) fi od;

for(n=1,1e7,if(denominator(k=sigma(n,-1))==1, print1(k", "))) \\ Charles R Greathouse IV, Mar 09 2014

a(n) = sigma(A007691(n))/A007691(n)

More terms from Jud McCranie, Jul 09 2000

More terms from David Wasserman, Jun 28 2004

A351551 Numbers k such that the largest unitary divisor of sigma(k) that is coprime with A003961(k) is also a unitary divisor of k.

Original entry on oeis.org

1, 2, 10, 34, 106, 120, 216, 260, 340, 408, 440, 580, 672, 696, 820, 1060, 1272, 1666, 1780, 1940, 2136, 2340, 2464, 3320, 3576, 3960, 4280, 4536, 5280, 5380, 5860, 6456, 6960, 7520, 8746, 8840, 9120, 9632, 10040, 10776, 12528, 12640, 13464, 14560, 16180, 16660, 17400, 17620, 19040, 19416, 19992, 21320, 22176, 22968

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

Numbers k for which A351546(k) is a unitary divisor of k.
The condition guarantees that A351555(k) = 0, therefore this is a subsequence of A351554.
The condition is also a necessary condition for A349745, therefore it is a subsequence of this sequence.
All six known 3-perfect numbers (A005820) are included in this sequence.
All 65 known 5-multiperfects (A046060) are included in this sequence.
Not all multiperfects (A007691) are present (only 587 of the first 1600 are), but all 23 known terms of A323653 are terms, while none of the (even) terms of A046061 or A336702 are.

			For n = 672 = 2^5 * 3^1 * 7^1, and the largest unitary divisor of the sigma(672) [= 2^5 * 3^2 * 7^1] coprime with A003961(672) [= 13365 = 3^5 * 5^1 * 11^1] is 2^5 * 7^1 = 224, therefore A351546(672) is a unitary divisor of 672, and 672 is included in this sequence.

Cf. A000203, A000396, A003961, A007691, A046061, A065997, A336702, A351546, A351555, A353633 (characteristic function).
Subsequence of A351552 and of A351554.
Cf. A349745, A351550 (subsequences), A005820, A046060, A323653 (very likely subsequences).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351546(n) = { my(f=factor(sigma(n)),u=A003961(n)); prod(k=1,#f~,f[k,1]^((0!=(u%f[k,1]))*f[k,2])); };
isA351551(n) =  { my(u=A351546(n)); (!(n%u) && 1==gcd(u,n/u)); };

A342924 Composite numbers k such that A003415(sigma(k)) = k + p*A003415(k), for some prime p, where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

Original entry on oeis.org

6, 28, 120, 496, 672, 963, 1036, 5871, 8128, 10479, 164284, 264768, 523776, 2308203, 6511664, 33550336, 41240261, 75384301, 400902412, 459818240, 581013140, 1253768516, 1476304896, 2114464203, 8589869056

Offset: 1

Antti Karttunen, Apr 08 2021

Composite numbers k for which A342926(k) = p*A003415(k), for some prime p.

Corresponding prime p for the first 25 terms is: 2, 2, 3, 2, 3, 3, 3, 11, 2, 11, 2, 3, 3, 5, 2, 2, 101, 397, 2, 3, 5, 7, 3, 5, 2. - Antti Karttunen, Feb 25 2022

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Odd terms in this sequence form a subsequence of A347884.
Cf. A342925, A342926.
Cf. A000396, A005820, A046060, A065997 (subsequences).
Cf. also A342922, A342923, A007691.

Block[{f}, f[n_] := If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Select[Range[4, 10^6], And[CompositeQ[#], PrimeQ[(f[DivisorSigma[1, #]] - #)/f[#] ]] &]] (* Michael De Vlieger, Apr 08 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342925(n) = A003415(sigma(n));
isA342924(n) = if((n<2)||isprime(n),0,my(q=(A342925(n)-n)/A003415(n)); ((1==denominator(q))&&isprime(q)));

Terms a(21) - a(25) from Antti Karttunen, Feb 25 2022

A083865 Sums of (one or more distinct) k-perfect numbers.

Original entry on oeis.org

6, 28, 34, 120, 126, 148, 154, 496, 502, 524, 530, 616, 622, 644, 650, 672, 678, 700, 706, 792, 798, 820, 826, 1168, 1174, 1196, 1202, 1288, 1294, 1316, 1322, 8128, 8134, 8156, 8162, 8248, 8254, 8276, 8282, 8624, 8630, 8652, 8658, 8744, 8750, 8772, 8778

Offset: 1

Torsten Klar (klar(AT)radbruch.jura.uni-mainz.de), Jun 18 2003

Each k-perfect number (A007691\{1}) appears once, and may also appear at most once in each sum of k-perfect numbers to create other terms in the sequence. [Harvey P. Dale, Feb 07 2012]

			a(3) = 34 because it is the sum of 6 + 28 both of which are perfect numbers.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A065997, A000695.

N:= 10000: # to get all terms <= N
Kperf:= select(t -> numtheory:-sigma(t) mod t = 0, [$2..N]):
S:= {0}:
for k in Kperf do S:= S union (k +~ S) od:
sort(convert(S minus {0}, list)); # Robert Israel, Nov 29 2016

With[{perf=Select[Range[10000],DivisorSigma[1,#]==2#&]},Rest[Union[Total/@ Subsets[perf]]]] (* Harvey P. Dale, Feb 07 2012 *)

a=[];n=1;until(50<#a=concat(a,vector(#a+1,i,n+if(i>1,a[i-1]))),while(sigma(n++)%n,));a  \\ M. F. Hasler, Feb 09 2012

Empirical observation: a(n) = 2*n + Sum_{k >= 1} 4^k*floor(2*n/2^k) for 1 <= n <= 15 and 32 <= n <= 47; a(n) = 2*n - 1344 + Sum_{k >= 1} 4^k*floor(2*n/2^k) for 16 <= n <= 31. Note 1344 = 4^3 + 4^4 + 4^5. Cf. A000695. - Peter Bala, Nov 29 2016

If b(n) = 2*n + Sum_{k >= 1} 4^k*floor(2*n/2^k) - a(n), we also have b(n) = 1344 for 48 <= n <= 63, then 2400 for 64 <= n <= 79, 3744 for 80 <= n <= 95, 8008 for 96 <= n <= 111, etc. The first case where b(n) is not constant on an interval 16*k <= n <= 16*k+15 is k=57214, where b(915431)=2747770287196 but b(915432)=2747770287312. - Robert Israel, Nov 29 2016

Corrected by M. F. Hasler and others, Feb 07 2012

A219545 Integer values of sigma(n)/n that are prime.

Original entry on oeis.org

2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2, 5, 5, 3, 2, 5, 5, 5, 5, 5, 5, 2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 2, 5, 7, 2, 5, 5, 7, 5, 5, 5, 7, 7, 7, 7, 5, 5, 7, 7, 7, 7, 5, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Jonathan Sondow, Nov 22 2012

nonn

Subsequence of A054030 consisting of primes among the abundancies sigma(m)/m of multiply perfect numbers m (see A007691).
Each 2 corresponds to a perfect number A000396, so if there are infinitely many perfect numbers, then the sequence is infinite.
If, in addition, there are only finitely many multiply perfect numbers m with sigma(m)/m > 2 (see A134639), then a(n) = 2 for all n > some N.

			A065997(1) = 6 and sigma(6)/6 = (1+2+3+6)/6 = 2, so a(1) = 2.

R. K. Guy, Unsolved Problems in Number Theory, B2.

A. Flammenkamp, The Multiply Perfect Numbers Page

Cf. A000396, A007691, A054030, A065997, A134639.

Select[Table[DivisorSigma[1,n]/n,{n,10^6}],PrimeQ] (* The program only generates the first seven terms of the sequence. To generate them all, the value of n would have to be greatly increased. *) (* Harvey P. Dale, Oct 25 2021 *)

a(n) = sigma(A065997(n))/A065997(n).

Extended by T. D. Noe, Nov 27 2012

A359168 Numbers k for which there is an odd number of prime factors (when counted with multiplicity) in k*sigma(k), where sigma is the sum of divisors function.

Original entry on oeis.org

3, 4, 5, 6, 8, 9, 10, 13, 16, 18, 21, 23, 25, 26, 27, 28, 33, 35, 37, 42, 44, 46, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 63, 64, 66, 68, 70, 73, 74, 76, 83, 85, 87, 88, 89, 91, 93, 95, 96, 99, 102, 103, 106, 110, 112, 114, 116, 118, 120, 122, 123, 124, 125, 126, 129, 131, 136, 139, 141, 143, 145, 146

Offset: 1

Antti Karttunen, Dec 19 2022

nonn

Numbers k such that the parities of A001222(k) and A058063(k) differ.

Positions of -1's in A359166.
Cf. A000203, A001222, A058063, A359167 (complement).
Cf. also A358767, A358768.
Cf. A000396, A005820, A046060, A065997 (subsequences).

isA359168(n) = ((bigomega(n)+bigomega(sigma(n)))%2);

A347884 Odd composites k for which A003415(sigma(k))-k is strictly positive and a multiple of A003415(k). Here A003415 is the arithmetic derivative.

Original entry on oeis.org

963, 969, 5871, 10479, 2308203, 41240261, 52024391, 69989429, 75384301, 319255721, 634457761, 781718149, 1184197307, 1190942957, 1195786661, 2114464203

Offset: 1

Antti Karttunen, Sep 19 2021

Odd nonprimes k for which A343223(k) = A003415(k).
Any odd terms of A065997, including odd perfect numbers, odd triperfect numbers and odd 5-multiperfect numbers, should occur in this sequence, if such numbers exist at all.
The odd terms present in A342924 form a subsequence of this sequence.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203, A003415, A007691, A065997, A342925, A342926, A343223.
Subsequence of A343218.
Cf. also A000396, A005820, A046060, A342922, A342923, A342924, A347872, A347880.

ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[1, 2.5*10^6, 2], CompositeQ[#] && (d = ad[DivisorSigma[1, #]] - #) > 0 && Divisible[d, ad[#]] &] (* Amiram Eldar, Sep 19 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA347884(n) = if(!(n%2)||isprime(n),0,my(u=(A003415(sigma(n))-(n))); ((u>0)&&!(u%A003415(n))));

A361469 a(n) = bigomega(A249670(A003961(n))).

Original entry on oeis.org

0, 3, 3, 3, 4, 4, 4, 7, 3, 7, 3, 4, 4, 5, 7, 6, 4, 6, 5, 7, 7, 6, 4, 6, 4, 5, 7, 5, 6, 8, 3, 9, 6, 7, 8, 6, 4, 6, 7, 11, 4, 8, 6, 4, 7, 5, 5, 7, 4, 5, 5, 3, 5, 8, 5, 9, 8, 9, 3, 8, 4, 6, 7, 7, 8, 7, 6, 7, 5, 9, 3, 8, 6, 5, 7, 6, 7, 8, 5, 10, 6, 7, 5, 6, 8, 7, 9, 10, 4, 10, 8, 5, 6, 6, 9, 10, 4, 7, 6, 5, 5, 6, 6, 7, 11

Offset: 1

Antti Karttunen, Mar 20 2023

nonn

Conjecture: There are no 1's in this sequence. If true, it would imply that there are no odd terms in A065997.

The first n with a(n) = 2 is 1684804. Note that A003961(1684804) = 5659641 is so far the only known odd term in A247086.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A001222, A003961, A065997, A247086, A249670, A361468.

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A249670(n) = { my(ab = sigma(n)/n); numerator(ab)*denominator(ab); };
A361469(n) = bigomega(A249670(A003961(n)));

a(n) = A001222(A361468(n)) = A001222(A249670(A003961(n))).

cp's OEIS Frontend

A000396 Perfect numbers k: k is equal to the sum of the proper divisors of k.

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

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A054030 Sigma(n)/n for n such that sigma(n) is divisible by n.

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A351551 Numbers k such that the largest unitary divisor of sigma(k) that is coprime with A003961(k) is also a unitary divisor of k.

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A342924 Composite numbers k such that A003415(sigma(k)) = k + p*A003415(k), for some prime p, where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

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A083865 Sums of (one or more distinct) k-perfect numbers.

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A219545 Integer values of sigma(n)/n that are prime.