A326051 -id:A326051 - 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.

A019283 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,6)-perfect numbers.

Original entry on oeis.org

42, 84, 160, 336, 1344, 86016, 550095, 1376256, 5505024, 22548578304

Offset: 1

N. J. A. Sloane

If 2^p-1 is a Mersenne prime then m = 21*2^(p-1) is in the sequence. Because sigma(sigma(m)) = sigma(sigma(21*2^(p-1))) = sigma(32*(2^p-1)) = 63*2^p = 6*(21*2^(p-1)) = 6*m. So 21*(A000668+1)/2 is a subsequence of this sequence. This is the subsequence 42, 84, 336, 1344, 86016, 1376256, 5505024, 22548578304, 24211351596743786496, ... - Farideh Firoozbakht, Dec 05 2005
See also the Cohen-te Riele links under A019276.
No other terms < 5 * 10^11. - Jud McCranie, Feb 08 2012
Any odd perfect numbers must occur in this sequence, as such numbers must be in the intersection of A000396 and A326051, that is, satisfy both sigma(n) = 2n and sigma(2n) = 6n, thus in combination they must satisfy sigma(sigma(n)) = 6n. Note that any odd perfect number should occur also in A326181. - Antti Karttunen, Jun 16 2019
a(11) > 4*10^12. - Giovanni Resta, Feb 26 2020

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Index entries for sequences where any odd perfect numbers must occur

Cf. A000668, A019278, A019279, A019282.

Cf. A000203, A000396, A005820, A051027, A326051, A326181.

Do[If[DivisorSigma[1, DivisorSigma[1, n]]==6n, Print[n]], {n, 6000000}] (* Farideh Firoozbakht, Dec 05 2005 *)

isok(n) = sigma(sigma(n))/n  == 6; \\ Michel Marcus, May 12 2016

a(10) by Jud McCranie, Feb 08 2012

A326181 Numbers n for which sigma(sigma(n)) = 3*sigma(n).

Original entry on oeis.org

54, 56, 87, 95, 276, 308, 429, 446, 455, 501, 581, 611, 158928, 194928, 195072, 199950, 226352, 234608, 236432, 248325, 255678, 263504, 266192, 273050, 275415, 304575, 336903, 341162, 353675, 366575, 369425, 369843, 380463, 386313, 389463, 406565, 411725, 415925, 422303, 447587, 468743, 497333, 500993, 511829, 515267, 519557, 519677

Offset: 1

Antti Karttunen, Jun 16 2019

nonn

Any odd perfect numbers must occur in this sequence, as such numbers must be in the intersection of A000396 and A326051, that is, satisfy both sigma(n) = 2n and sigma(2n) = 6n = 3*2n, thus in combination they must satisfy sigma(sigma(n)) = 3*sigma(n). Note that odd perfect numbers should occur also in A019283.

If, as conjectured, A005820 has 6 terms, then this sequence is finite and has 756 terms. - Giovanni Resta, Jun 17 2019

Giovanni Resta, Table of n, a(n) for n = 1..756 (first 169 terms from Antti Karttunen)
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A000396, A005820, A019283, A051027, A087943, A272027 (3*sigma(n)), A326051.

Subsequence of A066961.

isA326181(n) = { my(s=sigma(n)); (sigma(s)==3*s); };

A341622 Numbers that are either already perfect, or a perfect number is eventually reached if we start doubling them.

Original entry on oeis.org

3, 6, 7, 14, 28, 31, 62, 124, 127, 248, 254, 496, 508, 1016, 2032, 4064, 8128, 8191, 16382, 32764, 65528, 131056, 131071, 262112, 262142, 524224, 524284, 524287, 1048448, 1048568, 1048574, 2096896, 2097136, 2097148, 4193792, 4194272, 4194296, 8387584, 8388544, 8388592, 16775168, 16777088, 16777184, 33550336, 33554176, 33554368

Offset: 1

Antti Karttunen, Feb 19 2021

nonn

Numbers whose closure under map x -> 2x contains a perfect number (one of the terms of A000396).
Numbers k such that A341621(k) > A336915(k). No powers of 2 are included because they stay deficient forever.
Sequence is the union of odd perfect numbers (whose existence is contested, see e.g., A326051), and the numbers of the form (2^p - 1) * 2^e, where p is one of the primes in A000043, and e < p.

Cf. A000043, A000396, A005101, A326051, A336915, A336921.

Subsequence of A335431 provided there are no odd perfect numbers.

m = MersennePrimeExponent[Range[8]]; f[p_] := 2^Range[0, p - 1]*(2^p - 1); Select[Sort @ Flatten[f /@ m], # <= 2^m[[-1]] - 1 &] (* Amiram Eldar, Feb 20 2021, for calculating terms below 10^1500, the current lower bound for odd perfect numbers *)

isA341622(n) = if(!bitand(n,n-1), 0, for(i=0,oo,my(n2 = n+n); if(sigma(n) >= n2, return(sigma(n)==n2)); n = n2));

A351548 a(n) = A349745(n) divided by 2 if it is even, and 0 if A349745(n) is odd.

Original entry on oeis.org

0, 60, 108, 336, 1232, 11088, 114240, 261888, 320320, 418880, 2790720, 2882880, 3769920, 6499584, 9801792, 16930368, 19171152, 35672000, 47736000, 51068160, 98654400, 110046720, 172540368, 229909120, 403504640, 487788480, 738152448, 755415680, 886792320, 1960686000, 2070484416, 2339064000, 2889432000

Offset: 1

Antti Karttunen, Feb 18 2022

nonn

Questions: Are all nonzero terms abundant (in A005101)? Are all terms even? Could either be proved? See also comments in A351538 and in A351549.

The terms a(2) .. a(52) are all also practical (A005153) and Zumkeller (A083207). - Antti Karttunen, Dec 05 2024

Antti Karttunen, Table of n, a(n) for n = 1..52 (prepared from the b-file of A349745 computed by _Amiram Eldar_)
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A005101, A005153, A083207, A326051 (all six known terms are present here), A329963, A349169, A349745, A351458, A351459, A351538.

Cf. also A351549.

A351548(n) = { my(u=A349745(n)); if(u%2,0,u/2); };

a(n) = 0 if A349745(n) is odd, a(n) = A349745(n)/2 otherwise.

A219544 Even multiply-perfect numbers divided by 2.

Original entry on oeis.org

3, 14, 60, 248, 336, 4064, 15120, 16380, 261888, 1089270, 11784960, 16775168, 22766400, 71495424, 229909120, 689727360, 738152448, 4294934528, 7091219520, 15999197760, 21930739200, 25500590080, 33216860160

Offset: 1

Jonathan Sondow, Nov 22 2012

nonn

Just as no odd perfect number is known, all known multiply-perfect numbers A007691 greater than 1 are even.
The subsequence of even perfect numbers divided by 2 is A133028.
The subsequence of even triperfects / 2 is A326051. - Antti Karttunen, Mar 20 2021

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

Cf. A007691, A133028, A194771, A326051.

a(n) = A007691(n+1)/2.

a(n) = A194771(n+1)/4. [provided there are no odd multiperfect numbers > 1] - Antti Karttunen, Mar 20 2021

A341623 Numbers k such that sigma(3k) = 8k.

Original entry on oeis.org

28, 90, 496, 546, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176

Offset: 1

Antti Karttunen, Feb 19 2021

Every perfect number P greater than 6 (so, P is not divisible by 3) will be found in this sequence. Proof: sigma(3*P) = sigma(3)*sigma(P) = 4*(2*P) = 8*P. - Timothy L. Tiffin, Aug 26 2021

Solutions are integers y/3 where sigma(y)/y = 8/3. - Michel Marcus, Aug 27 2021

			546 is a term, since sigma(3*546) = sigma(1638) = 4368 = 8*546. - _Timothy L. Tiffin_, Aug 26 2021

Cf. A000396 (subsequence, apart from its terms that are divisible by 3).
Subsequence of A005101 and A227303.
Cf. also A000203, A144613, A326051, A336702, A347261.

Select[Range[5*10^9], DivisorSigma[1, 3*#] == 8*# &] (* Timothy L. Tiffin, Aug 26 2021 *)
Do[If[DivisorSigma[1, 3*k] == 8*k, Print[k]], {k, 5*10^9}] (* Timothy L. Tiffin, Aug 26 2021 *)

a(7)-a(8) from Martin Ehrenstein, Mar 06 2021

a(9)-a(10) from Michel Marcus, Aug 27 2021

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.

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A019283 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,6)-perfect numbers.

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A326181 Numbers n for which sigma(sigma(n)) = 3*sigma(n).

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A341622 Numbers that are either already perfect, or a perfect number is eventually reached if we start doubling them.

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A351548 a(n) = A349745(n) divided by 2 if it is even, and 0 if A349745(n) is odd.

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A219544 Even multiply-perfect numbers divided by 2.

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A341623 Numbers k such that sigma(3*k) = 8*k.

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A341623 Numbers k such that sigma(3k) = 8k.