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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• more

Comments

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)

Examples

			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
		

References

• 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.

Links

• 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)

Crossrefs

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.

Programs

Maple

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

Mathematica

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 *)

PARI

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

Formula

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

Extensions

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.