%I M5376 #160 Jul 03 2023 14:56:49

%S 120,672,523776,459818240,1476304896,51001180160

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

%C These six terms are believed to comprise all 3-perfect numbers. - cf. the MathWorld link. - _Daniel Forgues_, May 11 2010

%C 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

%C 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

%C 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

%C From _Antti Karttunen_, Mar 20 2021, Sep 18 2021, (Start):

%C 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).

%C 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.

%C (End)

%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 120, p. 42, Ellipses, Paris 2008.

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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D I. Stewart, L'univers des nombres, "Les nombres multiparfaits", Chap.15, pp 82-5, Belin/Pour la Science, Paris 2000.

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

%H Abiodun E. Adeyemi, <a href="https://arxiv.org/abs/1906.05798">A Study of @-numbers</a>, arXiv:1906.05798 [math.NT], 2019.

%H K. A. Broughan and Qizhi Zhou, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Broughan/broughan17d.html">Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4</a>, JIS 13 (2010) 10.1.5

%H A. Brousseau, <a href="http://www.fq.math.ca/numbertheory-tables.html">Number Theory Tables</a>, Fibonacci Association, San Jose, CA, 1973, p. 138.

%H S. Colbert-Pollack, J. Holdener, E. Rachfal, and Y. Xu <a href="https://doi.org/10.1080/10724117.2020.1849911">A DIY Project: Construct Your Own Multiply Perfect Number!</a>, Math Horizons, Vol. 28, pp. 20-23, February 2021.

%H F. Firoozbakht and M. F. Hasler, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Hasler/hasler2.html">Variations on Euclid's formula for Perfect Numbers</a>, JIS 13 (2010) #10.3.1.

%H Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/mpn.html">The Multiply Perfect Numbers Page</a> [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]

%H James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=DhPtIf-hpuU">The Six Triperfect Numbers</a>, Numberphile video (2018).

%H Fred Helenius, <a href="http://pw1.netcom.com/~fredh/index.html">Link to Glossary and Lists</a>

%H M. Kishore, <a href="https://doi.org/10.1090/S0025-5718-1984-0725999-4">Odd Triperfect Numbers</a>, Mathematics of Computation, vol. 42, no. 165, 1984, pp. 231-233.

%H G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiperfect">Multiperfect and hemiperfect numbers</a>

%H Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">Abundancy : Some Resources</a>

%H N. J. A. Sloane & A. L. Brown, <a href="/A005820/a005820.pdf">Correspondence, 1974</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MultiperfectNumber.html">Multiperfect Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Sous-Double.html">Sous-Double</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Multiply_perfect_number#Triperfect_numbers">Multiply perfect number</a>, (section Triperfect numbers)

%F a(n) = 2*A326051(n). [provided no odd triperfect numbers exist] - _Antti Karttunen_, Jun 13 2019

%e 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

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

%t 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 *)

%t Select[Range[10^6],DivisorSigma[1,#]==3#&] (* _Harvey P. Dale_, Jul 03 2023 *)

%o (PARI) isok(n) = sigma(n, -1) == 3; \\ _Michel Marcus_, Nov 22 2015

%Y Cf. A000203, A000396, A007539, A017665, A019278, A027687, A046060, A046061, A068403, A075701, A097023, A171266, A259302, A259303, A306373, A326051, A326181, A329189, A335141, A335254, A347383, A347391.

%Y Subsequence of the following sequences: A007691, A069085, A153501, A216780, A292365, A336458, A336461, A336745, and if there are no odd terms, then also of A334410.

%Y Positions of 120's in A094759, 119's in A326200.

%K nonn,nice,more

%O 1,1

%A _N. J. A. Sloane_

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

%E Edited by _Farideh Firoozbakht_ and _N. J. A. Sloane_, Sep 09 2014 to remove some incorrect statements.