A228058 -id:A228058 - 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.
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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.
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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.
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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).
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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

A033879 Deficiency of n, or 2n - (sum of divisors of n).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 5, 2, 10, -4, 12, 4, 6, 1, 16, -3, 18, -2, 10, 8, 22, -12, 19, 10, 14, 0, 28, -12, 30, 1, 18, 14, 22, -19, 36, 16, 22, -10, 40, -12, 42, 4, 12, 20, 46, -28, 41, 7, 30, 6, 52, -12, 38, -8, 34, 26, 58, -48, 60, 28, 22, 1, 46, -12, 66, 10, 42, -4, 70, -51

Offset: 1

N. J. A. Sloane

Records for the sequence of the absolute values are in A075728 and the indices of these records in A074918. - R. J. Mathar, Mar 02 2007
a(n) = 1 iff n is a power of 2. a(n) = n - 1 iff n is prime. - Omar E. Pol, Jan 30 2014
If a(n) = 1 then n is called a least deficient number or an almost perfect number. All the powers of 2 are least deficient numbers but it is not known if there exists a least deficient number that is not a power of 2. See A000079. - Jianing Song, Oct 13 2019
It is not known whether there are any -1's in this sequence. See comment in A033880. - Antti Karttunen, Feb 02 2020

			For n = 10 the divisors of 10 are 1, 2, 5, 10, so the deficiency of 10 is 10 minus the sum of its proper divisors or simply 10 - 5 - 2 - 1 = 2. - _Omar E. Pol_, Dec 27 2013

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

N. J. A. Sloane, Table of n, a(n) for n = 1..25000 [First 2000 terms from T. D. Noe, terms up to 16384 from Antti Karttunen]
Nichole Davis, Dominic Klyve and Nicole Kraght, On the difference between an integer and the sum of its proper divisors, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493.
Jose A. B. Dris, Conditions Equivalent to the Descartes-Frenicle-Sorli Conjecture on Odd Perfect Numbers, arXiv preprint arXiv:1610.01868 [math.NT], 2016.
Jose Arnaldo B. Dris, Analysis of the Ratio D(n)/n, arXiv:1703.09077 [math.NT], 2017.
Jose Arnaldo Bebita Dris, On a curious biconditional involving the divisors of odd perfect numbers, Notes on Number Theory and Discrete Mathematics, 23(4) (2017), 1-13.
Jose Arnaldo Bebita Dris and Immanuel Tobias San Diego, Some Modular Considerations Regarding Odd Perfect Numbers, arXiv:2002.12139 [math.NT], 2020.
Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada, Conditions equivalent to the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers - Part II, Notes on Number Theory and Discrete Mathematics (2018) Vol. 24, No. 3, 62-67.
Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada, A note on the OEIS sequence A228059, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 1, 199-205.
Index entries for sequences related to sigma(n).

Cf. A000396 (positions of zeros), A005100 (of positive terms), A005101 (of negative terms).
Cf. A000203, A033880, A074918, A075728, A192895, A286385, A286449, A295881, A295882.
Cf. A083254 (Möbius transform), A228058, A296074, A296075, A323910, A325636, A325826, A325970, A325976.
Cf. A141545 (positions of a(n) = -12).
For this sequence applied to various permutations of natural numbers and some other sequences, see A323174, A323244, A324055, A324185, A324546, A324574, A324575, A324654, A325379.

with(numtheory): A033879:=n->2*n-sigma(n): seq(A033879(n), n=1..100);

Table[2n-DivisorSigma[1,n],{n,80}] (* Harvey P. Dale, Oct 24 2011 *)

a(n)=2*n-sigma(n) \\ Charles R Greathouse IV, Oct 13 2016

from sympy import divisor_sigma
def A033879(n): return (n<<1)-divisor_sigma(n) # Chai Wah Wu, Apr 13 2024

[2*n-sigma(n, 1) for n in range(1, 73)] # Stefano Spezia, Jul 18 2025

a(n) = -A033880(n).
a(n) = A005843(n) - A000203(n). - Omar E. Pol, Dec 14 2008
a(n) = n - A001065(n). - Omar E. Pol, Dec 27 2013
G.f.: 2*x/(1 - x)^2 - Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 24 2017
a(n) = A286385(n) - A252748(n). - Antti Karttunen, May 13 2017
From Antti Karttunen, Dec 29 2017: (Start)
a(n) = Sum_{d|n} A083254(d).
a(n) = Sum_{d|n} A008683(n/d)*A296075(d).
a(n) = A065620(A295881(n)) = A117966(A295882(n)).
a(n) = A294898(n) + A000120(n).
(End)
From Antti Karttunen, Jun 03 2019: (Start)
Sequence can be represented in arbitrarily many ways as a difference of the form (n - f(n)) - (g(n) - n), where f and g are any two sequences whose sum f(n)+g(n) = sigma(n). Here are few examples:
a(n) = A325314(n) - A325313(n) = A325814(n) - A034460(n) = A325978(n) - A325977(n).
a(n) = A325976(n) - A325826(n) = A325959(n) - A325969(n) = A003958(n) - A324044(n).
a(n) = A326049(n) - A326050(n) = A326055(n) - A326054(n) = A326044(n) - A326045(n).
a(n) = A326058(n) - A326059(n) = A326068(n) - A326067(n).
a(n) = A326128(n) - A326127(n) = A066503(n) - A326143(n).
a(n) = A318878(n) - A318879(n).
a(A228058(n)) = A325379(n). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - Pi^2/12 = 0.177532... . - Amiram Eldar, Dec 07 2023

Definition corrected by N. J. A. Sloane, Jul 04 2005

A191218 Odd numbers n such that sigma(n) is congruent to 2 modulo 4.

Original entry on oeis.org

5, 13, 17, 29, 37, 41, 45, 53, 61, 73, 89, 97, 101, 109, 113, 117, 137, 149, 153, 157, 173, 181, 193, 197, 229, 233, 241, 245, 257, 261, 269, 277, 281, 293, 313, 317, 325, 333, 337, 349, 353, 369, 373, 389, 397, 401, 405, 409, 421, 425, 433, 449, 457, 461, 477

Offset: 1

Luis H. Gallardo, May 26 2011

Exactly the numbers of the form p^{4k+1}*m^2 with p a prime congruent to 1 modulo 4 and m a positive integer coprime with p. The odd perfect numbers are all of this form.

See A228058 for the terms where m > 1. - Antti Karttunen, Apr 22 2019

			For n=3 one has a(3)=17 since sigma(17) = 18 = 4*4 +2 is congruent to 2 modulo 4

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Subsequence of A191217.
Cf. A228058, A324898 (subsequences).
Cf. A000203, A191219, A324647, A324718, A324719.

with(numtheory): genodd := proc(b) local n,s,d; for n from 1 to b by 2 do s := sigma(n);
if modp(s,4)=2 then print(n); fi; od; end;

Select[Range[1,501,2],Mod[DivisorSigma[1,#],4]==2&] (* Harvey P. Dale, Nov 12 2017 *)

forstep(n=1,10^3,2,if(2==(sigma(n)%4),print1(n,", "))) \\ Joerg Arndt, May 27 2011

list(lim)=my(v=List()); forstep(e=1,logint(lim\=1,5),4, forprimestep(p=5,sqrtnint(lim,e),4, my(pe=p^e); forstep(m=1,sqrtint(lim\pe),2, if(m%p, listput(v,pe*m^2))))); Set(v) \\ Charles R Greathouse IV, Feb 16 2022

A336547 Numbers k such that for 1 <= i < j <= h, all sigma(p_i^e_i), sigma(p_j^e_j) are pairwise coprime, when k = p_1^e_1 * ... * p_h^e_h, where each p_i^e_i is the maximal power of prime p_i dividing k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 31, 32, 36, 37, 38, 41, 43, 44, 45, 47, 48, 49, 50, 53, 54, 56, 59, 61, 62, 63, 64, 67, 68, 71, 72, 73, 74, 75, 76, 79, 80, 81, 83, 86, 89, 92, 96, 97, 99, 100, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 121, 122, 124, 125

Offset: 1

Antti Karttunen, Jul 25 2020

nonn

Numbers k such that A051027(k) = Product_{p^e||k} A051027(p^e) = A353802(n). Here each p^e is the maximal prime power divisor of k, and A051027(k) = sigma(sigma(k)). Numbers at which points A051027 appears to be multiplicative.
Proof that this interpretation is equal to the main definition:
  (1) If none of sigma(p_1^e_1), ..., sigma(p_k^e_k) share prime factors, then A051027(k) = sigma(sigma(p_1^e_1) * ... * sigma(p_k^e_k)) = A051027(p_1^e_1) * ... * A051027(p_k^e_k), by multiplicativity of sigma.
  (2) On the other hand, if say, gcd(sigma(p_i^e_i), sigma(p_j^e_j)) = c > 1 for some distinct i, j, then that c has at least one prime factor q, with product t = sigma(p_1^e_1) * ... * sigma(p_k^e_k) having a divisor of the form q^v (where v = valuation(t,q)), and the same prime factor q occurs as a divisor in more than one of the sigma(p_i^e_j), in the form q^k, with the exponents summing to v, then it is impossible to form sigma(q^v) = (1 + q + q^2 + ... + q^v) as a product of some sigma(q^k_1) * ... * sigma(q^k_z), i.e., as a product of (1 + q + ... + q^k_1) * ... * (1 + q + ... + q^k_z), with v = k_1 + ... + k_z, because such a product is always larger than (1 + q + ... + q^v). And if there are more such cases of "split primes", then each of them brings its own share to this monotonic inequivalence, thus Product_{p^e|n} A051027(p^e) = A353802(n) >= A051027(n), for all n.
From Antti Karttunen, May 07 2022: (Start)
Also numbers k such that A062401(k) = phi(sigma(n)) = Product_{p^e||k} A062401(p^e) = A353752(n).
Proof that also this interpretation is equal to the main definition:
  (1) like in (1) above, if none of sigma(p_1^e_1), ..., sigma(p_k^e_k) share prime factors, then by the multiplicativity of phi.
  (2) On the other hand, if say, gcd(sigma(p_i^e_i), sigma(p_j^e_j)) = c > 1 for some distinct i, j, then that c must have a prime factor q occurring in both sigma(p_i^e_i) and sigma(p_j^e_j), with say q^x being the highest power of q in the former, and q^y in the latter. Then phi(q^x)*phi(q^y) < phi(q^(x+y)), i.e. here the inequivalence acts to the opposite direction than with sigma(sigma(...)), so we have A353752(n) <= A062401(n) for all n.
(End)
From Antti Karttunen, May 22 2022: (Start)
All even perfect numbers (even terms of A000396) are included in this sequence. In general, for any perfect number n in this sequence, map k -> A026741(sigma(k)) induces on its unitary prime power divisors (p^e||n) a permutation that is a single cycle, mapping each one of them to the next larger one, except that the largest is mapped to the smallest one. Therefore, for a hypothetical odd perfect number n = x*a*b*c*d*e*f*g*h to be included in this sequence, where x is Euler's special factor of the form (4k+1)^(4h+1), and a .. h are even powers of odd primes (of which there are at least eight distinct ones, see P. P. Nielsen reference in A228058), further constraints are imposed on it: (1) that h < x < 2*a (here assuming that a, b, c, ..., g, h have already been sorted by their size, thus we have a < b < ... < g < h < x < 2*a, and (2), that we must also have sigma(a) = b, sigma(b) = c, ..., sigma(f) = g, sigma(h) = x, and sigma(x) = 2*a. Note that of the 400 initial terms of A008848, only its second term 81 is a prime power, so empirically this seems highly unlikely to ever happen.
(End)

			28 = 2^2 * 7 is present, as sigma(2^2) = 7 and sigma(7) = 8, and 7 and 8 are relatively prime (do not share prime factors). Likewise for all even terms of A000396. - _Antti Karttunen_, May 09 2022

Antti Karttunen, Table of n, a(n) for n = 1..25000
Index entries for sequences related to sigma(n)

Cf. A051027, A062401, A336546 (characteristic function), A336548 (complement).
Positions of zeros in A336562, in A353753, and in A353803.
Positions of ones in A353755, in A353784, and in A353806.
Union of A000961 and A336549.
Subsequence of A336358 and of A336557.
Cf. also A000396, A008848, A228058, A231484, A353752, A353802.

PARI
```
isA336547(n) = A336546(n);
```

{k | A336562(k) == 0}. - Antti Karttunen, May 09 2022

The old definition moved to comments and replaced with an alternative definition from the comment section by Antti Karttunen, May 07 2022

A349169 Numbers k such that k * gcd(sigma(k), A003961(k)) is equal to the odd part of {sigma(k) * gcd(k, A003961(k))}, where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 15, 105, 3003, 3465, 13923, 45045, 264537, 459459, 745875, 1541475, 5221125, 8729721, 10790325, 14171625, 29288025, 34563375, 57034575, 71430975, 99201375, 109643625, 144729585, 205016175, 255835125, 295708875, 356080725, 399242025, 419159475, 449323875, 928602675, 939495375, 1083656925, 1941623775, 1962350685, 2083228875

Offset: 1

Antti Karttunen, Nov 10 2021

Numbers k such that A348990(k) [= k/gcd(k, A003961(k))] is equal to A348992(k), which is the odd part of A349162(k), thus all terms must be odd, as A348990 preserves the parity of its argument.
Equally, numbers k for which gcd(A064987(k), A191002(k)) is equal to A000265(gcd(A064987(k), A341529(k))).
Also odd numbers k for which A348993(k) = A319627(k).
Odd terms of A336702 are given by the intersection of this sequence and A349174.
Conjectures:
  (1) After 1, all terms are multiples of 3. (Why?)
  (2) After 1, all terms are in A104210, in other words, for all n > 1, gcd(a(n), A003961(a(n))) > 1. Note that if we encountered a term k with gcd(k, A003961(k)) = 1, then we would have discovered an odd multiperfect number.
  (3) Apart from 1, 15, 105, 3003, 13923, 264537, all other terms are abundant. [These apparently are also the only terms that are not Zumkeller, A083207. Note added Dec 05 2024]
  (4) After 1, all terms are in A248150. (Cf. also A386430).
  (5) After 1, all terms are in A348748.
  (6) Apart from 1, there are no common terms with A349753.
Note: If any of the last four conjectures could be proved, it would refute the existence of odd perfect numbers at once. Note that it seems that gcd(sigma(k), A003961(k)) < k, for all k except these four: 1, 2, 20, 160.
Questions:
  (1) For any term x here, can 2*x be in A349745? (Partial answer: at least x should be in A191218 and should not be a multiple of 3). Would this then imply that x is an odd perfect number? (Which could explain the points (1) and (4) in above, assuming the nonexistence of opn's).

Cf. A000203, A003961, A007949, A064989, A083207, A104210, A161942, A191002, A191218, A228058, A319627, A322361, A326134, A329963, A342671, A348748, A348990, A348992, A348993, A349162, A349164, A349174.
Cf. also A349745, A349746, A349753.
Subsequence of A349749, and of A386430.

Select[Range[10^6], #1/GCD[#1, #3] == #2/(2^IntegerExponent[#2, 2]*GCD[#2, #3]) & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* Michael De Vlieger, Nov 11 2021 *)

A000265(n) = (n >> valuation(n, 2));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349169(n) = { my(s=sigma(n),u=A003961(n)); (n*gcd(s,u) == A000265(s)*gcd(n,u)); }; \\ (Program simplified Nov 30 2021)

For all n >= 1, A007949(A000203(a(n))) = A007949(a(n)). [sigma preserves the 3-adic valuation of the terms of this sequence] - Antti Karttunen, Nov 29 2021

Name changed and comment section rewritten by Antti Karttunen, Nov 29 2021

A325314 a(n) = n - A162296(n), where A162296(n) is the sum of divisors of n that have a square factor.

Original entry on oeis.org

1, 2, 3, 0, 5, 6, 7, -4, 0, 10, 11, -4, 13, 14, 15, -12, 17, -9, 19, -4, 21, 22, 23, -24, 0, 26, -9, -4, 29, 30, 31, -28, 33, 34, 35, -43, 37, 38, 39, -32, 41, 42, 43, -4, -9, 46, 47, -64, 0, -25, 51, -4, 53, -54, 55, -40, 57, 58, 59, -36, 61, 62, -9, -60, 65, 66, 67, -4, 69, 70, 71, -111, 73, 74, -25, -4, 77, 78, 79, -88, -36, 82, 83

Offset: 1

Antti Karttunen, Apr 21 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A013661, A033879, A162296, A228058, A325313, A325315, A325320.

Array[# - DivisorSigma[1, #] + Times @@ (1 + FactorInteger[#][[;; , 1]]) - Boole[# == 1] &, 83] (* Michael De Vlieger, Jun 06 2019, after Amiram Eldar at A162296 *)

A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));

a(n) = n - A162296(n).
a(n) = A033879(n) + A325313(n).
a(A228058(n)) = -A325320(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - zeta(2)/2 = 0.1775329665... . - Amiram Eldar, Feb 22 2024

A325385 a(n) = gcd(n-A048250(n), n-A162296(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 5, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 19, 2, 1, 4, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 41, 1, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 1, 3, 1, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, Apr 24 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A048250, A126795, A162296, A228058, A325313, A325314, A325375.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325385(n) = gcd(n-A048250(n),n-A162296(n));

a(n) = gcd(n-A048250(n), n-A162296(n)).
a(n) = gcd(A325313(n), A325314(n)).
a(A228058(n)) = A325375(n).

A324213 Number of k with 0 <= k <= sigma(n) such that n-k and 2n-sigma(n) are relatively prime.

Original entry on oeis.org

2, 4, 3, 8, 4, 2, 4, 16, 12, 9, 6, 14, 6, 12, 8, 32, 10, 26, 8, 21, 14, 18, 12, 20, 30, 16, 18, 2, 14, 24, 10, 64, 16, 24, 22, 88, 14, 30, 26, 36, 18, 32, 14, 42, 26, 28, 24, 54, 56, 80, 20, 32, 26, 40, 36, 60, 38, 42, 30, 56, 18, 42, 48, 128, 42, 48, 22, 50, 28, 72, 26, 122, 26, 54, 58, 46, 48, 56, 26, 86, 120, 60, 42, 96, 54

Offset: 1

Antti Karttunen and David A. Corneth, May 26 2019, with better name from Charlie Neder, Jun 02 2019

nonn

Number of ways to form the sum sigma(n) = x+y so that n-x and n-y are coprime, with x and y in range 0..sigma(n).
From Antti Karttunen, May 28 - Jun 08 2019: (Start)
Empirically, it seems that a(n) >= A034444(n) and also that a(n) >= A034444(A000203(n)) unless n is in A000396.
Specifically, if it could be proved that a(n) >= A034444(n)/2 for n >= 2, which in turn would imply that a(n) >= A001221(n) for all n, then we would know that no odd perfect numbers could exist. Note that a(n) must be 2 on all perfect numbers, whether even or odd. See also A325819.
(End)

			For n=1, sigma(1) = 1, both gcd(1-0, 1-(1-0)) = gcd(1,0) = 1 and gcd(1-1, 1-(1-1)) = gcd(0,1) = 1, thus a(1) = 2.
--
For n=3, sigma(3) = 4, we have 5 cases to consider:
  gcd(3-0, 3-(4-0)) = 1 = gcd(3-4, 3-(4-4)),
  gcd(3-1, 3-(4-1)) = 2 = gcd(3-3, 3-(4-3)),
  gcd(3-2, 3-(4-2)) = 1,
of which three cases give 1 as a result, thus a(3) = 3.
--
For n=6, sigma(6) = 12, we have 13 cases to consider:
  gcd(6-0, 6-(12-0)) = 6 = gcd(6-12, 6-(12-12)),
  gcd(6-1, 6-(12-1)) = 5 = gcd(6-11, 6-(12-11)),
  gcd(6-2, 6-(12-2)) = 4 = gcd(6-10, 6-(12-10)),
  gcd(6-3, 6-(12-3)) = 3 = gcd(6-9, 6-(12-9)),
  gcd(6-4, 6-(12-4)) = 2 = gcd(6-8, 6-(12-8))
  gcd(6-5, 6-(12-5)) = 1 = gcd(6-7, 6-(12-7)),
  gcd(6-6, 6-(12-6)) = 0,
of which only two give 1 as a result, thus a(6) = 2.
--
For n=10, sigma(10) = 18, we have 19 cases to consider:
  gcd(10-0, 10-(18-0)) = 2 = gcd(10-18, 10-(18-18)),
  gcd(10-1, 10-(18-1)) = 1 = gcd(10-17, 10-(18-17)),
  gcd(10-2, 10-(18-2)) = 2 = gcd(10-16, 10-(18-16)),
  gcd(10-3, 10-(18-3)) = 1 = gcd(10-15, 10-(18-15)),
  gcd(10-4, 10-(18-4)) = 2 = gcd(10-14, 10-(18-14)),
  gcd(10-5, 10-(18-5)) = 1 = gcd(10-13, 10-(18-13)),
  gcd(10-6, 10-(18-6)) = 2 = gcd(10-12, 10-(18-12)),
  gcd(10-7, 10-(18-7)) = 1 = gcd(10-11, 10-(18-11)),
  gcd(10-8, 10-(18-8)) = 2 = gcd(10-10, 10-(18-10)),
  gcd(10-9, 10-(18-9)) = 1,
of which 9 cases give 1 as a result, thus a(10) = 9.
--
For n=15, sigma(15) = 24, we have 25 cases to consider:
  gcd(15-0, 15-(24-0)) = 3 = gcd(15-24, 15-(24-24)),
  gcd(15-1, 15-(24-1)) = 2 = gcd(15-23, 15-(24-23)),
  gcd(15-2, 15-(24-2)) = 1 = gcd(15-22, 15-(24-22)),
  gcd(15-3, 15-(24-3)) = 6 = gcd(15-21, 15-(24-21)),
  gcd(15-4, 15-(24-4)) = 1 = gcd(15-20, 15-(24-20)),
  gcd(15-5, 15-(24-5)) = 2 = gcd(15-19, 15-(24-19)),
  gcd(15-6, 15-(24-6)) = 3 = gcd(15-18, 15-(24-18)),
  gcd(15-7, 15-(24-7)) = 2 = gcd(15-17, 15-(24-17)),
  gcd(15-8, 15-(24-8)) = 1 = gcd(15-16, 15-(24-16)),
  gcd(15-9, 15-(24-9)) = 6 = gcd(15-15, 15-(24-15)),
  gcd(15-10, 15-(24-10)) = 1 = gcd(15-14, 15-(24-14)),
  gcd(15-11, 15-(24-11)) = 2 = gcd(15-13, 15-(24-13)),
  gcd(15-12, 15-(24-12)) = 3,
of which 2*4 = 8 cases give 1 as a result, thus a(15) = 8.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000010, A000203, A000396, A001221, A014567, A034444, A058062, A062401, A228058, A325807, A325815, A325816, A325817, A325818, A325819, A325960, A325961, A325962, A325965, A325966, A325967, A325968, A325970, A325971, A325972.

Array[Sum[Boole[1 == GCD[#1 - i, #1 - (#2 - i)]], {i, 0, #2}] & @@ {#, DivisorSigma[1, #]} &, 85] (* Michael De Vlieger, Jun 09 2019 *)

A324213(n) = { my(s=sigma(n)); sum(i=0,s,(1==gcd(n-i,n-(s-i)))); };

a(n) = Sum_{i=0..sigma(n)} [1 == gcd(n-i,n-(sigma(n)-i))], where [ ] is the Iverson bracket and sigma(n) is A000203(n).
a(A000396(n)) = 2.
a(n) = A325815(n) + A034444(n).
a(n) = 1+A000203(n) - A325816(n).
a(A228058(n)) = A325819(n).

cp's OEIS Frontend

A387167 Numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) = 1, and that satisfy Euler's condition for odd perfect numbers (A228058).

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A387404 Numbers of the form 12*k + 1 that satisfy Euler's condition for odd perfect numbers (A228058).

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

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A033879 Deficiency of n, or 2n - (sum of divisors of n).

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A191218 Odd numbers n such that sigma(n) is congruent to 2 modulo 4.

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A336547 Numbers k such that for 1 <= i < j <= h, all sigma(p_i^e_i), sigma(p_j^e_j) are pairwise coprime, when k = p_1^e_1 * ... * p_h^e_h, where each p_i^e_i is the maximal power of prime p_i dividing k.

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A349169 Numbers k such that k * gcd(sigma(k), A003961(k)) is equal to the odd part of {sigma(k) * gcd(k, A003961(k))}, where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

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