A259302 -id:A259302 - 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.

A259303 Numbers n such that there exists an x!=n that makes {x,x,n} an amicable multiset.

Original entry on oeis.org

1560, 7380, 20448, 20664, 22752, 23352, 25248, 26964, 35328, 51360, 100200, 104412, 108792, 135540, 156312, 192096, 232740, 252252, 278460, 310752, 362160, 401688, 492660, 543456, 548592, 588744, 659736, 732060, 767520, 826848, 998712, 999648, 1129440, 1129464

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

We call the multiset {x,x,n} amicable iff sigma(x)=sigma(n)=x+x+n.
For the x values, see A259302.
If the condition x!=n were dropped, the terms from A005820 would also belong here.

John Cerkan, Table of n, a(n) for n = 1..1742

Cf. A005820, A259307, A259302, A259304, A259305, A259306, A125490.

isOK(n) = x=(sigma(n)-n)/2; x>0 && x!=n && denominator(x)==1 && sigma(x)==x+x+n;

A259304 Numbers n such that there exists an x!=n that makes {n,n,n,x} an amicable multiset.

Original entry on oeis.org

796320, 1205820, 1277640, 1963080, 5154240, 8599500, 8694000, 9022860, 10820880, 12149280, 15845760, 17173800, 18023040, 19323360, 21360240, 22720320, 25494480, 27052200, 27335880, 28365480, 28574280, 28704060, 29211000, 30329640, 30373200, 30450420, 31433220

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

We call the multiset {n,n,n,x} amicable iff sigma(n)=sigma(x)=n+n+n+x.
For the x values, see A259305.
If the condition x!=n were dropped, the terms from A027687 would also belong here.

John Cerkan, Table of n, a(n) for n = 1..1000

Cf. A027687, A259307, A259305, A259302, A259303, A259306, A036471.

isOK(n) = x=sigma(n)-3*n; x>0&x!=n&sigma(x)==3*n+x

A259305 Numbers n such that there exists an x!=n that makes {x,x,x,n} an amicable multiset.

Original entry on oeis.org

756000, 1220940, 1437480, 1973160, 5019840, 9058140, 9703260, 11058480, 11641680, 13085280, 17089920, 18117000, 19370880, 21282912, 21628080, 25295760, 25585560, 26936640, 27763560, 28191240, 29104200, 29203020, 30450420, 32839560, 33279120, 33398820

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

We call the multiset {x,x,x,n} amicable iff sigma(x)=sigma(n)=x+x+x+n.
For the x values, see A259304.
If the condition x!=n were dropped, the terms from A027687 would also belong here.

John Cerkan, Table of n, a(n) for n = 1..1000

Cf. A027687, A259307, A259304, A259302, A259303, A259306, A036471.

isOK(n) = x=(sigma(n)-n)/3; x>0&x!=n&denominator(x)==1&sigma(x)==3*x+n

A259306 Numbers n such that there exists an x!=n that makes {n,n,x,x} an amicable multiset.

Original entry on oeis.org

1441440, 1860768, 3205440, 3378240, 3423420, 3914820, 4029480, 4437720, 5738040, 6093360, 6807240, 7136640, 7239120, 7551360, 9402120, 10204740, 12270720, 12405120, 12942720, 13495680, 14627340, 14725620, 15233400, 17166240, 18097800, 19158048, 23023440

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

We call the multiset {n,n,x,x} amicable iff sigma(n)=sigma(x)=n+n+x+x.

If the condition x!=n were dropped, the terms from A027687 would also belong here.

John Cerkan, Table of n, a(n) for n = 1..1000

Cf. A027687, A259307, A259302, A259303, A259304, A259305, A036471.

isOK(n) = x=(sigma(n)-2*n)/2; x>0 && x!=n && denominator(x)==1 && sigma(x)==n+n+x+x

A259307 Numbers that belong to at least one amicable multiset.

Original entry on oeis.org

1, 6, 28, 120, 220, 284, 496, 672, 1184, 1210, 1560, 1740, 1980, 2016, 2556, 2620, 2924, 5020, 5564, 6232, 6368, 7380, 7776, 8128, 9180, 9504, 10744, 10856, 11556, 12285, 14595, 17296, 18416, 19260, 20448, 20640, 20664, 21168, 21384, 21924, 22200, 22428, 22752

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

Call a finite multiset {x_1, x_2, ..., x_k} of natural numbers (the x_i need not be distinct) an amicable multiset iff sigma(x_1)=sigma(x_2)=...=sigma(x_k)=x_1+x_2+...+x_k.
By definition, A255215 is a subset because a set can be regarded as a special multiset.
Also A007691 is a subset, since a k-perfect number corresponds to an amicable multiset in an obvious way. For example, since 120 is 3-perfect, the multiset {120, 120, 120} is amicable.
The first amicable multiset that belongs to neither A255215 nor A007691 is {1740, 1740, 1560}.

Jeppe Stig Nielsen, List of all amicable multisets with a sigma value below 10^7.

Cf. A255215, A007691, A259302, A259303, A259304, A259305, A259306.

/* write amicable multisets */ sMax=10^7;sigmaVals=vector(sMax,x,[]);for(n=1,sMax,s=sigma(n);s<=sMax&sigmaVals[s]=concat(sigmaVals[s],[n]));(MultisetSum(numbers,desiredSum,track)=if(desiredSum<0,return);if(desiredSum==0,print(apply(x->numbers[x],track));return);for(i=if(track,track[#track],1),#numbers,MultisetSum(numbers,desiredSum-numbers[i],concat(track,[i]))));for(s=1,sMax,MultisetSum(sigmaVals[s],s,[]))

A273969 Numbers n such that there exists a pair x,y, where x

Original entry on oeis.org

702240, 817740, 1156680, 1159200, 1811040, 2450448, 2570400, 2784600, 3534300, 3912480, 4228560, 4546080, 4702320, 5682600, 6902280, 7280280, 7469280, 7706160, 8225280, 8316000, 8465184, 8522640, 8639400, 9025380, 9256800, 9282000, 9492120, 9828000

Offset: 1

John Cerkan, Jul 17 2016

nonn

We call the multiset {x,y,n,n} amicable iff sigma(x) = sigma(y) = sigma(n) = x+y+n+n. For the x values, see A273970. For the y values, see A273971.

If the condition xA259306 would also belong here.

			sigma(695520) = sigma(803040) = sigma(702240) = 695520 + 803040 + 702240 + 702240.

John Cerkan, Table of n, a(n) for n = 1..8082
John Cerkan, Python code

Cf. A259302, A259303, A259304, A259305, A259306, A273970, A273971.

A273970 Numbers x such that there exist a pair y, n with x < y, x != n and y != n that makes {x,y,n,n} an amicable multiset.

Original entry on oeis.org

695520, 753480, 1113840, 1136520, 1784160, 2313360, 2898720, 3140280, 3865680, 3960600, 4272840, 4500720, 4626720, 6126120, 6167700, 7197960, 7442820, 7731360, 8177400, 8498700, 8784720, 8828820, 8920800, 8966160, 9124920, 9232860, 9664200, 9729720

Offset: 1

John Cerkan, Jul 17 2016

nonn

We call the multiset {x,y,n,n} amicable iff sigma(x) = sigma(y) = sigma(n) = x+y+n+n. For the n values, see A273969. For the y values, see A273971.

If the condition xA259306 would also belong here.

			sigma(695520) = sigma(803040) = sigma(702240) = 695520 + 803040 + 702240 + 702240.

John Cerkan, Table of n, a(n) for n = 1..8060
John Cerkan, Python code

Cf. A259302, A259303, A259304, A259305, A259306, A273969, A273971.

A273971 Numbers y such that there exists a pair x, n, with x < y, x != n and y != n that makes {x,y,n,n} an amicable multiset.

Original entry on oeis.org

756000, 803040, 1267560, 1442448, 1851360, 2535120, 3209760, 3477240, 3926160, 3969840, 4413240, 4664880, 6094368, 6840540, 7617960, 7783020, 8027880, 8360352, 8586900, 9215640, 9559200, 9596520, 9697380, 9811620, 9815400, 9938160, 10063200, 10234224

Offset: 1

John Cerkan, Jul 17 2016

nonn

We call the multiset {x,y,n,n} amicable iff sigma(x) = sigma(y) = sigma(n) = x+y+n+n. For the n values, see A273969. For the x values, see A273970.

If the condition xA259306 would also belong here.

			sigma(695520) = sigma(803040) = sigma(702240) = 695520 + 803040 + 702240 + 702240.

John Cerkan, Table of n, a(n) for n = 1..7515
John Cerkan, Python code

Cf. A259302, A259303, A259304, A259305, A259306, A273969, A273970.

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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A259303 Numbers n such that there exists an x!=n that makes {x,x,n} an amicable multiset.

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A259304 Numbers n such that there exists an x!=n that makes {n,n,n,x} an amicable multiset.

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A259305 Numbers n such that there exists an x!=n that makes {x,x,x,n} an amicable multiset.

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A259306 Numbers n such that there exists an x!=n that makes {n,n,x,x} an amicable multiset.

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A259307 Numbers that belong to at least one amicable multiset.

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A273969 Numbers n such that there exists a pair x,y, where x

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A273970 Numbers x such that there exist a pair y, n with x < y, x != n and y != n that makes {x,y,n,n} an amicable multiset.

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A273971 Numbers y such that there exists a pair x, n, with x < y, x != n and y != n that makes {x,y,n,n} an amicable multiset.

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