A036471 -id:A036471 - cp's OEIS frontend

A273928 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

53542288800, 67154527440, 67509842400, 87113426400, 87502615200, 86133247200, 86133247200, 86133247200, 98471252880

Offset: 1

John Cerkan, Jun 04 2016

The 5-tuple starting with 53542288800 was given by Donovan Johnson. The common value of sigma(x) is 294821130240.
A larger 5-tuple, (55766707476480, 56992185169920, 57515254917120, 57754372515840, 57829096765440), was found by Michel Marcus on Dec 09 2013. The common value of sigma(x) is 285857616844800.
A still larger example (227491164588441600, 228507506351308800, 229862628701798400, 230878970464665600, 243752632794316800), probably the first one to be published, had been found by Yasutoshi Kohmoto in 2008, cf. link to SeqFan post.
Other terms from John Cerkan.
There are different definitions for amicable k-tuples, cf. link to MathWorld.

John Cerkan, More terms with gaps.
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Nov 23 2008
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Dec 09 2013
Eric W. Weisstein, Amicable Triple. From MathWorld--A Wolfram Web Resource.

Cf. A233553, A273930, A273931, A273933, A273934, A273936 (5-tuples).
Cf. A036471 - A036474 and A116148 (quadruples).
Cf. A125490 - A125492 and A137231 (triples).

A273930 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

Original entry on oeis.org

59509850400, 68763895200, 72747675000, 88410722400, 88021533600, 89894684880, 89894684880, 90391981680, 102481394400

Offset: 1

John Cerkan, Jun 04 2016

The 5-tuple starting with 53542288800 was given by Donovan Johnson. The common value of sigma(x) is 294821130240.
A larger 5-tuple, (55766707476480, 56992185169920, 57515254917120, 57754372515840, 57829096765440), was found by Michel Marcus on Dec 09 2013. The common value of sigma(x) is 285857616844800.
A still larger example (227491164588441600, 228507506351308800, 229862628701798400, 230878970464665600, 243752632794316800), probably the first one to be published, had been found by Yasutoshi Kohmoto in 2008, cf. link to SeqFan post.
Other terms from John Cerkan.
There are different definitions for amicable k-tuples, cf. link to MathWorld.

John Cerkan, More terms, with gaps.
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Nov 23 2008
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Dec 09 2013
Eric W. Weisstein, Amicable Triple. From MathWorld--A Wolfram Web Resource.

Cf. A233553, A273928, A273931, A273933, A273934, A273936 (5-tuples).
Cf. A036471 - A036474 and A116148 (quadruples).
Cf. A125490 - A125492 and A137231 (triples).

A273931 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

Original entry on oeis.org

59999219280, 69626138400, 73605331800, 89398663200, 89398663200, 90391981680, 94320626400, 94832992800, 103169959200

Offset: 1

John Cerkan, Jun 04 2016

The 5-tuple starting with 53542288800 was given by Donovan Johnson. The common value of sigma(x) is 294821130240.
A larger 5-tuple, (55766707476480, 56992185169920, 57515254917120, 57754372515840, 57829096765440), was found by Michel Marcus on Dec 09 2013. The common value of sigma(x) is 285857616844800.
A still larger example (227491164588441600, 228507506351308800, 229862628701798400, 230878970464665600, 243752632794316800), probably the first one to be published, had been found by Yasutoshi Kohmoto in 2008, cf. link to SeqFan post.
Other terms from John Cerkan.
There are different definitions for amicable k-tuples, cf. link to MathWorld.

John Cerkan, More terms, with gaps.
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Nov 23 2008
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Dec 09 2013
Eric Weisstein's World of Mathematics, Amicable Triple.

Cf. A233553, A273928, A273930, A273933, A273934, A273936 (5-tuples).
Cf. A036471 - A036474 and A116148 (quadruples).
Cf. A125490 - A125492 and A137231 (triples).

A273933 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

Original entry on oeis.org

60074174160, 71957405520, 75710489400, 96058282320, 96058282320, 97306569360, 96759542880, 94972878000, 109117562400

Offset: 1

John Cerkan, Jun 04 2016

The 5-tuple starting with 53542288800 was given by Donovan Johnson. The common value of sigma(x) is 294821130240.
A larger 5-tuple, (55766707476480, 56992185169920, 57515254917120, 57754372515840, 57829096765440), was found by Michel Marcus on Dec 09 2013. The common value of sigma(x) is 285857616844800.
A still larger example (227491164588441600, 228507506351308800, 229862628701798400, 230878970464665600, 243752632794316800), probably the first one to be published, had been found by Yasutoshi Kohmoto in 2008, cf. link to SeqFan post.
Other terms from John Cerkan.
There are different definitions for amicable k-tuples, cf. link to MathWorld.

John Cerkan, More terms, with gaps.
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Nov 23 2008
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Dec 09 2013
Eric W. Weisstein, Amicable Triple. From MathWorld--A Wolfram Web Resource.

Cf. A233553, A273928, A273930, A273931, A273934, A273936 (5-tuples).
Cf. A036471 - A036474 and A116148 (quadruples).
Cf. A125490 - A125492 and A137231 (triples).

A273934 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

Original entry on oeis.org

61695597600, 72598125600, 78953074200, 96369633360, 96369633360, 103073639760, 99692021520, 100469023200, 109446377040

Offset: 1

John Cerkan, Jun 04 2016

The 5-tuple starting with 53542288800 was given by Donovan Johnson. The common value of sigma(x) is 294821130240.
A larger 5-tuple, (55766707476480, 56992185169920, 57515254917120, 57754372515840, 57829096765440), was found by Michel Marcus on Dec 09 2013. The common value of sigma(x) is 285857616844800.
A still larger example (227491164588441600, 228507506351308800, 229862628701798400, 230878970464665600, 243752632794316800), probably the first one to be published, had been found by Yasutoshi Kohmoto in 2008, cf. link to SeqFan post.
Other terms from John Cerkan.
There are different definitions for amicable k-tuples, cf. link to MathWorld.

John Cerkan, More terms, with gaps.
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Nov 23 2008
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Dec 09 2013
Eric W. Weisstein, Amicable Triple. From MathWorld--A Wolfram Web Resource.

Cf. A233553, A273928, A273930, A273931, A273933, A273936 (5-tuples).
Cf. A036471 - A036474 and A116148 (quadruples).
Cf. A125490 - A125492 and A137231 (triples).

A273936 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

Original entry on oeis.org

294821130240, 350100092160, 368526412800, 457350727680, 457350727680, 466800122880, 466800122880, 466800122880, 522686545920

Offset: 1

John Cerkan, Jun 04 2016

The 5-tuple starting with 53542288800 was given by Donovan Johnson. The common value of sigma(x) is 294821130240.
A larger 5-tuple, (55766707476480, 56992185169920, 57515254917120, 57754372515840, 57829096765440), was found by Michel Marcus on Dec 09 2013. The common value of sigma(x) is 285857616844800.
A still larger example (227491164588441600, 228507506351308800, 229862628701798400, 230878970464665600, 243752632794316800), probably the first one to be published, had been found by Yasutoshi Kohmoto in 2008, cf. link to SeqFan post.
Other terms from John Cerkan.
There are different definitions for amicable k-tuples, cf. link to MathWorld.

John Cerkan, More terms, with gaps.
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Nov 23 2008
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Dec 09 2013
Eric W. Weisstein, Amicable Triple. From MathWorld--A Wolfram Web Resource.

Cf. A233553, A273928, A273930, A273931, A273933, A273934 (5-tuples).
Cf. A036471 - A036474 and A116148 (quadruples).
Cf. A125490 - A125492 and A137231 (triples).

A036470 a(n) is the number of distinct possible values of d(k), the number of divisors of k, among numbers k whose binary order (A029837) does not exceed n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 11, 12, 16, 17, 23, 26, 31, 37, 43, 48, 58, 64, 74, 82, 94, 106, 122, 133, 146, 165, 183, 202, 224, 244, 267, 294, 325, 355, 389, 416, 453, 500, 541, 584, 636, 680, 737, 795, 859, 922, 995, 1068, 1149, 1233, 1324, 1412, 1523, 1616, 1731, 1845

Offset: 0

Labos Elemer

nonn

			If 1 <= k <= 128, i.e., the binary order of k is g(k) <= 7, then d(k) takes 12 values {1,2,3,4,5,6,7,8,9,10,12,16}; thus a(7) = 12. The maximal value (16) appears as a(7) in A036451.

David A. Corneth, Table of n, a(n) for n = 0..63

Cf. A000005, A005179, A029837, A036471, A036493.

a(20)-a(21) corrected by David A. Corneth, May 12 2018

A255215 Numbers that belong to at least one amicable tuple.

Original entry on oeis.org

1, 220, 284, 1184, 1210, 1980, 2016, 2556, 2620, 2924, 5020, 5564, 6232, 6368, 9180, 9504, 10744, 10856, 11556, 12285, 14595, 17296, 18416, 21168, 22200, 23940, 27312, 31284, 32136, 37380, 38940, 39480, 40068, 40608, 41412, 41952, 42168, 43890, 46368, 47124

Offset: 1

Jeppe Stig Nielsen, Feb 17 2015

nonn

Call a finite set {x_1, x_2, ..., x_k} of natural numbers (the x_i are pairwise distinct) an amicable k-tuple iff sigma(x_1)=sigma(x_2)=...=sigma(x_k)=x_1+x_2+...+x_k. Here sigma=A000203. For k=1, the only possible amicable one-tuple is {1}. For k=2 we get the classical amicable pairs (A063990). k=3 is amicable triples (A125490), k=4 amicable quadruples (A036471), and so on. A natural number n belongs to this sequence if and only if n is a member of some amicable k-tuple.
By definition, this sequence contains no duplicates.
For k<>2, an amicable k-tuple is not an aliquot cycle.

			1 belongs to this sequence because {1} is considered an amicable one-tuple.
284 belongs to this sequence because {220, 284} is an amicable pair.
2016 belongs to this sequence because {1980, 2016, 2556} is an amicable triple.
38940 is included in this sequence only once even if both {38940, 40068, 41952} and {38940, 40608, 41412} are amicable.
1000 does not belong to this sequence. To prove that, note that sigma(1000)=2340. Then find all x such that sigma(x)=2340, these are 792, 1000, 1062, 1305, 1611, 1945, 2339. Run through all subsets of 792, 1000, 1062, 1305, 1611, 1945, 2339 that include 1000 to verify that no such subset has a sum of 2340.
A tuple (or multiset) like {1560, 1740, 1740} where some element(s) are repeated, is not allowed here, and neither 1560 nor 1740 belongs to this sequence.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..361

Cf. A000203, A063990, A125490, A036471.

Cf. A259307 (duplicates allowed in tuple).

(notSubsetSum(desiredSum, searchSet) = { /* strongly inspired by is_A006037 function from A006037 */ local(t); /* return nonzero iff desiredSum is not the sum of a subset of searchSet */ setsearch( Set(searchSet), desiredSum ) & return /* equal to one element of searchSet */; while( #searchSet & searchSet[ #searchSet]>desiredSum, searchSet=vecextract(searchSet, "^-1")); desiredSum >= (t = sum(i=1, #searchSet, searchSet[i])) & return( desiredSum-t /* nonzero if desiredSum>t */ ); desiredSum > searchSet[ #searchSet] & ! notSubsetSum( desiredSum - searchSet[ #searchSet], searchSet=vecextract( searchSet, "^-1" )) & return; notSubsetSum( desiredSum, searchSet ) }); (othersWithSameSigma(n) = { s=sigma(n); [ x | x<-[1..s-1] , sigma(x)==s&&x!=n ] }); (is_A255215(x) = !notSubsetSum(sigma(x)-x, othersWithSameSigma(x)))

A233626 Least member of an amicable n-tuple: (x[1],...,x[n]) such that sigma(x[1])=...=sigma(x[n])=x[1]+...+x[n], x[i]

Original entry on oeis.org

1, 220, 1980, 3270960, 53542288800

Offset: 1

M. F. Hasler, Dec 12 2013

Some authors use other definitions for amicable k-tuples, cf. link to MathWorld.

Eric W. Weisstein, Amicable Triple. From MathWorld--A Wolfram Web Resource.

Cf. A002025, A002046 and A161005 for amicable pairs.
Cf. A125490 - A125492 and A137231 for amicable triples.
Cf. A036471 - A036474 and A116148 for amicable quadruples.
Cf. A233553 for amicable quintuples.

A359334 Amicable numbers k that can be expressed as a sum k = x+y = A001065(x) + A001065(y) and a sum k = z+t = A001065(z) + A001065(t) where (x, y, z, t) are parts of two amicable pairs and A001065(i) is the sum of the aliquot parts of i.

Original entry on oeis.org

67212, 1296000, 20528640, 37739520, 75479040, 321408000, 348364800, 556839360, 579156480, 638668800, 661893120, 761177088, 796340160, 883872000, 1181174400, 1282417920, 2068416000, 2395008000, 2682408960, 3155023872, 3599769600, 4049740800, 4606156800, 4716601344

Offset: 1

Zoltan Galantai, Dec 26 2022

nonn

From Michel Marcus, Dec 31 2022: (Start)
In other words, numbers k that can be expressed as a sum k = x+y = z+t where either (x,y) and (z,t), or (x,z) and (y,t), are 2 amicable pairs.
Note that there is currently a single instance of the case (x,z) and (y,t), and this corresponds to the value 64 that appears twice in A066539.
The other terms correspond to values appearing at least twice in A180164.
There are instances where it can appear 3 times, and the least instance is 64795852800 for the 3 amicable pairs [29912035725, 34883817075], [31695652275, 33100200525], [32129958525, 32665894275].
There are instances where it can appear 6 times, and the least instance is 4169926656000 for the 6 amicable pairs [1953433861918, 2216492794082], [1968039941816, 2201886714184], [1981957651366, 2187969004634], [1993501042130, 2176425613870], [2046897812505, 2123028843495], [2068113162038, 2101813493962]. (End)

			67212 is a term because 67212 = 220 + 66992 = 284 + 66928 where (220, 284) and (66928, 66992) are two amicable pairs.
1296000 is a term because 1296000 = 609928 + 686072 = 643336 + 652664 where (609928, 686072) and (643336, 652664) are two amicable pairs.

Song Y. Yan, Perfect, Amicable and Sociable Numbers, World Scientific Pub Co Inc, 1996, pp. 113-121.

Michel Marcus, Table of n, a(n) for n = 1..233
Eric Weisstein's World of Mathematics, Amicable Pair

Cf. A002025, A063990, A259180, A259933, A036471, A180164, A001065, A066539.

More terms from Amiram Eldar, Dec 31 2022

cp's OEIS Frontend

A273928 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

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A273930 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

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A273931 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

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A273933 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

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A273934 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

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A273936 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

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A036470 a(n) is the number of distinct possible values of d(k), the number of divisors of k, among numbers k whose binary order (A029837) does not exceed n.

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A255215 Numbers that belong to at least one amicable tuple.

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A233626 Least member of an amicable n-tuple: (x[1],...,x[n]) such that sigma(x[1])=...=sigma(x[n])=x[1]+...+x[n], x[i]

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A359334 Amicable numbers k that can be expressed as a sum k = x+y = A001065(x) + A001065(y) and a sum k = z+t = A001065(z) + A001065(t) where (x, y, z, t) are parts of two amicable pairs and A001065(i) is the sum of the aliquot parts of i.

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