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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.
(Formerly M5376)
120, 672, 523776, 459818240, 1476304896, 51001180160 (list; graph; refs; listen; history; text; internal format)



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) = 3. - Michel Marcus, Nov 22 2015


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.

David Wells, "The Penguin Book of Curious and Interesting Numbers," Penguin Books, London, 1986, pages 135, 159 and 185.


Table of n, a(n) for n=1..6.

K. A. Broughan, Qizhi Zhou, Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4, JIS 13 (2010) 10.1.5

A. Brousseau, Number Theory Tables, Fibonacci Association, San Jose, CA, 1973, p. 138.

F. Firoozbakht, M. 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]

Fred Helenius, Link to Glossary and Lists

G. 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


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


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 *)


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


Cf. A000396, A007539, A027687, A046060, A046061.

Sequence in context: A114887 A069085 A039688 * A052787 A292970 A052769

Adjacent sequences:  A005817 A005818 A005819 * A005821 A005822 A005823




N. J. A. Sloane


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.



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Last modified October 21 20:41 EDT 2017. Contains 293707 sequences.