A069085 -id:A069085 - cp's OEIS frontend

A005820 3-perfect (triply perfect, tri-perfect, triperfect or sous-double) numbers: numbers such that the sum of the divisors of n is 3n.

120, 672, 523776, 459818240, 1476304896, 51001180160

Offset: 1

These six terms are believed to comprise all 3-perfect numbers. - cf. the MathWorld link. - Daniel Forgues, May 11 2010
If there exists an odd perfect number m (a famous open problem) then 2m would be 3-perfect, since sigma(2m) = sigma(2)*sigma(m) = 3*2m. - Jens Kruse Andersen, Jul 30 2014
According to the previous comment from Jens Kruse Andersen, proving that this sequence is complete would imply that there are no odd perfect numbers. - Farideh Firoozbakht, Sep 09 2014
If 2 were prepended to this sequence, then it would be the sequence of integers k such that numerator(sigma(k)/k) = A017665(k) = 3. - Michel Marcus, Nov 22 2015
From  Antti Karttunen, Mar 20 2021, Sep 18 2021, (Start):
Obviously, any odd triperfect numbers k, if they exist, have to be squares for the condition sigma(k) = 3*k to hold, as sigma(k) is odd only for k square or twice a square. The square root would then need to be a term of A097023, because in that case sigma(2*k) = 9*k. (See illustration in A347391).
Conversely to Jens Kruse Andersen's comment above, any 3-perfect number of the form 4k+2 would be twice an odd perfect number. See comment in A347870.
(End)

			120 = 2^3*3*5;  sigma(120) = (2^4-1)/1*(3^2-1)/2*(5^2-1)/4 = (15)*(4)*(6) = (3*5)*(2^2)*(2*3) = 2^3*3^2*5 = (3) * (2^3*3*5) = 3 * 120. - _Daniel Forgues_, May 09 2010

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 120, p. 42, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, B2.
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", Chap.15, pp 82-5, Belin/Pour la Science, Paris 2000.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 142.
David Wells, "The Penguin Book of Curious and Interesting Numbers," Penguin Books, London, 1986, pages 135, 159 and 185.

Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.
Kevin A. Broughan and Qizhi Zhou, Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4, JIS 13 (2010) 10.1.5
Alfred Brousseau, Number Theory Tables, Fibonacci Association, San Jose, CA, 1973, p. 138.
Seth Colbert-Pollack, Judy Holdener, Emily Rachfal, and Yanqi Xu, A DIY Project: Construct Your Own Multiply Perfect Number!, Math Horizons, Vol. 28, pp. 20-23, February 2021.
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 [This page contains a lot of useful information, but be careful, not all the statements are correct. For example, it appears to claim that the six terms of this sequence are known to be complete, which is not the case. - _N. J. A. Sloane_, Sep 10 2014]
James Grime and Brady Haran, The Six Triperfect Numbers, Numberphile video (2018).
Shyam Sunder Gupta, Perfect, Multiply Perfect, and Sociable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.
Fred Helenius, Link to Glossary and Lists
Masao Kishore, Odd Triperfect Numbers, Mathematics of Computation, vol. 42, no. 165, 1984, pp. 231-233.
Gérard P. Michon, Multiperfect and hemiperfect numbers
Walter Nissen, Abundancy : Some Resources
N. J. A. Sloane & A. L. Brown, Correspondence, 1974
Eric Weisstein's World of Mathematics, Multiperfect Number
Eric Weisstein's World of Mathematics, Sous-Double
Wikipedia, Multiply perfect number, (section Triperfect numbers)

Cf. A000203, A000396, A007539, A017665, A019278, A027687, A046060, A046061, A068403, A075701, A097023, A171266, A259302, A259303, A306373, A326051, A326181, A329189, A335141, A335254, A347383, A347391.
Subsequence of the following sequences: A007691, A069085, A153501, A216780, A292365, A336458, A336461, A336745, and if there are no odd terms, then also of A334410.
Positions of 120's in A094759, 119's in A326200.

A005820:=n->`if`(numtheory[sigma](n) = 3*n, n, NULL): seq(A005820(n), n=1..6*10^5); # Wesley Ivan Hurt, Oct 15 2017

triPerfectQ[n_] := DivisorSigma[1, n] == 3n; A005820 = {}; Do[If[triPerfectQ[n], AppendTo[A005820, n]], {n, 10^6}]; A005820 (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)
Select[Range[10^6],DivisorSigma[1,#]==3#&] (* Harvey P. Dale, Jul 03 2023 *)

isok(n) = sigma(n, -1) == 3; \\ Michel Marcus, Nov 22 2015

a(n) = 2*A326051(n). [provided no odd triperfect numbers exist] - Antti Karttunen, Jun 13 2019

Wells gives the 6th term as 31001180160, but this is an error.

Edited by Farideh Firoozbakht and N. J. A. Sloane, Sep 09 2014 to remove some incorrect statements.

A075701 a(1)=1, a(n+1)=sigma(a(n))-2*a(n).

Original entry on oeis.org

1, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6, 24, 12, 4, -1, 3, -2, 7, -6

Offset: 1

Benoit Cloitre, Oct 02 2002

sign

Taking any nonperfect number as initial value, does the map x->sigma(x)-2x lead to the cycle (-1,3,-2,7,-6,24,12,4) if during the iteration no perfect number is reached? Example: 124 -> -24 -> 108 -> 64 -> -1 -> 3 -> -2 -> 7 -> -6 -> 24 -> 12 -> 4 and the cycle (-1,3,-2,7,-6,24,12,4) is reached.

There appear to be lots of other cycles, for example the numbers in A005820 are cycles of length one. For longer cycles refer to the discussion in links. - Hans Havermann, Jul 21 2013

Peter Luschny, Open questions posed in OEIS #1
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A146556, A005820, A069085, A113285.

NestList[DivisorSigma[1, #]-2#&, 1, 94]  (* Peter Luschny, Jul 17 2013 *)
Join[{1},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{-1, 3, -2, 7, -6, 24, 12, 4},93]] (* Ray Chandler, Aug 25 2015 *)

Periodic with period (-1, 3, -2, 7, -6, 24, 12, 4) of length 8.

A371921 The number of iterations of the map x -> A033880(x) starting at n until the a nonpositive number is reached, or 0 if this does not happen.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Apr 12 2024

Analogous to A098007 with A033880(n) = sigma(n) - 2*n instead of A001065(n) = sigma(n) - n.

			a(n) = 0 if the iterations that start at n are entering a cycle. Examples of cycles are:
  1) Cycles of length 1: the triperfect numbers (A005820), 120, 672, 523776, ..., which are the fixed points of A033880. The triperfect numbers can be reached from other values of n, e.g., 276, 448, 486, 510, 702, ... .
  2) Cycles of length 2: the only known cycle is (45840, 51168) (see A069085). It can be reached from other values of n, e.g., 32130, 39420, 45480, 66300, ... .
  3) Cycles of length 3: the least cycle is (243732672, 271303776, 256786848). It is first reached from n = 107689320.
  4) Cycles of length 4: the least cycle is (65071776, 82842816, 89761152, 77260656). It can be reached from other values of n, e.g., 33623940, 41132280, 42825888, ... . The next cycle of length 4 is (985948800, 1381340160, 2183133696, 1489384608).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma), A001065, A033880, A098007.

Cf. A005101, A005820, A069085, A113285, A263837, A371920.

ab[n_] := Module[{k}, If[n < 1, 0, k = DivisorSigma[1, n] - 2*n; If[k < 1, 0, k]]]; a[n_] := Module[{s = NestWhileList[ab, n, UnsameQ, All]}, If[s[[-1]] == 0, Length[s] - 2, 0]]; Array[a, 120]

ab(n) = {my(k); if(n < 1, 0, k = sigma(n) - 2*n; if(k < 1, 0, k));}
a(n) = {my(t = 0); until(bittest(t, n = ab(n)), t += 1<M. F. Hasler at A098007

a(n) = 1 if and only if n is nonabundant (A263837).
If a(n) > 0 then:
  a(n) > 1 if n is abundant (A005101).
  a(n) > 2 if n is in A371920.

A069146 Numbers m such that m = sigma(abs(k)) - 3k, where k = sigma(m) - 3m.

Original entry on oeis.org

1248, 1596, 28272, 30240, 32760, 463296, 2178540, 12865770, 23569920, 30998250, 45532800, 142990848, 1379454720, 1912369152, 2623977450, 43861478400, 66433720320, 153003540480, 403031236608, 489622536192, 704575228896

Offset: 1

Jason Earls, Apr 08 2002

1.5*10^12 < a(22) <= 7834005405696. If 2^k-1 > 3 is a prime (A000023), then 2^(k-1)*3*19*(2^k-1) is a term. - Giovanni Resta, Dec 11 2019

			Let n = 1248. The sum of the divisors of n is 3528, so k = 3528 - 3*1248 = -216. The sum of the divisors of 216 is 600 and 600 - 3*(-216) = 1248, so 1248 is in the sequence.

Cf. A000203, A069085, A001065.

Select[Range[5*10^5], DivisorSigma[1, Abs[(k = DivisorSigma[1, #] - 3#)]] -3k == # &] (* Amiram Eldar, Dec 11 2019 *)

More terms from David Wasserman, Apr 15 2003
a(12)-a(15) from Amiram Eldar, Dec 11 2019
a(16)-a(21) from Giovanni Resta, Dec 11 2019

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A005820 3-perfect (triply perfect, tri-perfect, triperfect or sous-double) numbers: numbers such that the sum of the divisors of n is 3n.

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A075701 a(1)=1, a(n+1)=sigma(a(n))-2*a(n).

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A371921 The number of iterations of the map x -> A033880(x) starting at n until the a nonpositive number is reached, or 0 if this does not happen.

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A069146 Numbers m such that m = sigma(abs(k)) - 3k, where k = sigma(m) - 3m.

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