A259180 -id:A259180 - cp's OEIS frontend

A162884 Half the difference between the larger and smaller term of the n-th amicable pair.

32, 13, 152, 272, 68, 56, 1155, 560, 6532, 32, 2025, 9009, 4490, 11835, 8775, 392, 5756, 13210, 2240, 2288, 9032, 2860, 42272, 40652, 55426, 21592, 8944, 8575, 5840, 1755, 34648, 38072, 33536, 38296, 4664, 57796, 35296, 30555, 10856, 41384, 88965, 22496

Offset: 1

Juri-Stepan Gerasimov, Jul 16 2009

nonn

			a(7)=1155 because the 7th pair of amicable numbers is 12285 and 14595; and (14595-12285)/2=1155.

Cf. A063990, A066539, A259180.

read("transforms3") ; L002046 := BFILETOLIST("b002046.txt") : L002025 := BFILETOLIST("b002025.txt") : A066539 := proc(n) global L002046,L002025; op(n,L002046)-op(n,L002025) ; end:
A162884 := proc(n) A066539(n)/2 ; end: seq(A162884(n),n=1..100) ; # R. J. Mathar, Jul 19 2009

With[{s = PositionIndex@ Array[DivisorSigma[1, #] &, 10^6]}, Flatten@ Map[Differences, Apply[Join, Map[Function[n, Select[Subsets[Lookup[s, n], {2}], Total@ # == n &]], Sort@ Select[Keys@ s, Length@ Lookup[s, #] > 1 &]]]]/2] (* Michael De Vlieger, Oct 26 2017 *)

a(n) = A066539(n)/2.

a(n) = (A259180(2n) - A259180(2n-1))/2. - Omar E. Pol, Oct 26 2017

Terms resorted along with A066539 by R. J. Mathar, Jul 19 2009

A275315 Average of amicable pairs (x,y), ordered by the smaller value x given in A002025.

Original entry on oeis.org

252, 1197, 2772, 5292, 6300, 10800, 13440, 17856, 69552, 66960, 69120, 78624, 84240, 112320, 131040, 122760, 147420, 155520, 174096, 178560, 194400, 199584, 322812, 349272, 374976, 378000, 446400, 477603, 508896, 524160, 635040, 648000, 657720, 673920, 648000, 725760, 761400, 833280, 890568, 939600

Offset: 1

Timothy L. Tiffin, Jul 22 2016

nonn

Each term represents the midpoint of an interval (x,y), where x (A002025) and y (A002046) form a pair of amicable numbers (A259180).  The length and radius of each interval can be found in A066539 and A162884, respectively.
This sequence is not monotonic (specifically, not nondecreasing), since x+y (A180164) is not monotonic.  For a monotonic (nondecreasing) ordering of these averages, see A275316.
It is unknown if there exists an amicable pair where x and y have opposite parity (one is even and the other is odd).  If such a pair is ever found, then the compound adjective "same-parity" will need to be added to the name of this sequence.

			a(  1) = (     220 +      284)/2 =      504/2 =      252.
a(  2) = (    1184 +     1210)/2 =     2394/2 =     1197.
a(  3) = (    2620 +     2924)/2 =     5544/2 =     2772.
...      ...                 ...          ...         ...
a(  9) = (   63020 +    76084)/2 =   139104/2 =    69552.
a( 10) = (   66928 +    66992)/2 =   133920/2 =    66960.
a( 11) = (   67095 +    71145)/2 =   138240/2 =    69120.
...      ...                 ...          ...         ...
a( 15) = (  122265 +   139815)/2 =   262080/2 =   131040.
a( 16) = (  122368 +   123152)/2 =   245520/2 =   122760.
a( 17) = (  141664 +   153176)/2 =   294840/2 =   147420.
...      ...                 ...          ...         ...
a( 32) = (  609928 +   686072)/2 =  1296000/2 =   648000.
...      ...                 ...          ...         ...
a( 35) = (  643336 +   652664)/2 =  1296000/2 =   648000.
...      ...                 ...          ...         ...
a(105) = ( 9478910 + 11049730)/2 = 20528640/2 = 10264320.
...      ...                 ...          ...         ...
a(109) = (10254970 + 10273670)/2 = 20528640/2 = 10264320.
...      ...                 ...          ...         ...
a(137) = (17754165 + 19985355)/2 = 37739520/2 = 18869760.
a(138) = (17844255 + 19895265)/2 = 37739520/2 = 18869760.
...      ...                 ...          ...         ...

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..142 from Timothy L. Tiffin)
VaxaSoftware, List of amicable numbers from 1 to 20,000,000 [142 pairs].

Cf. A002025, A002046, A066539, A162884, A180164, A259180, A275316.

a(n) = [A002025(n) + A002046(n)]/2 = A180164(n)/2.

A275469 Difference between the larger and smaller terms of the n-th amicable pair (x,y) given in A259933.

Original entry on oeis.org

64, 26, 304, 544, 136, 112, 2310, 1120, 64, 4050, 13064, 18018, 8980, 23670, 784, 17550, 11512, 26420, 4480, 4576, 18064, 5720, 84544, 81304, 110852, 43184, 17888, 17150, 11680, 3510, 69296, 76144, 9328, 67072, 76592, 115592, 70592, 61110, 21712, 82768

Offset: 1

Timothy L. Tiffin, Jul 28 2016

nonn

Each term represents the length of an interval (x,y), where x (A260086) and y (A260087) form a pair of amicable numbers (A259933).  The midpoint and radius of each interval can be found in A275316 and A275470, respectively.
Each term will be even as long as there does not exist an amicable pair where x and y have opposite parity.
This sequence is a rearrangement of A066539 (which is based on A002025, A002046, and A259180).  The first ten indices for which a(n) does not equal A066539(n) are n = 9, 10, 11, 15, 16, 33, 34, 35, 41, 42.

			a(1) = 284 - 220 = 64, a(2) = 1210 - 1184 = 26, and a(3) = 2924 - 2620 = 304.

Timothy L. Tiffin, Table of n, a(n) for n = 1..142
VaxaSoftware, List of amicable numbers from 1 to 20,000,000 [142 pairs].

Cf. A002025, A002046, A066539, A259180, A259933, A260086, A260087, A275316, A275470.

a(n) = A260087(n) - A260086(n).

A287026 List of pairs of amicable numbers (m, n) with record value of m/n.

Original entry on oeis.org

220, 284, 1184, 1210, 6232, 6368, 10744, 10856, 66928, 66992, 16137628, 16150628, 28118032, 28128368, 766292835, 766512285, 1930301618, 1930741582, 4000783984, 4001351168, 44303273584, 44307987968, 85617896265, 85625175735, 161899964416, 161905117184

Offset: 1

Amiram Eldar, Aug 31 2017

The 2 members in each pair are adjacent to each other in the sequence.

The pairs were found using Sergei Chernykh's Amicable pair list. There are 21 pairs with lesser member < 10^18.

			The ratios m/n are
                   220/284                 = 0.7746...
                  1184/1210                = 0.9785...
                  6232/6368                = 0.9786...
                 10744/10856               = 0.9896...
                 66928/66992               = 0.9990...
              16137628/16150628            = 0.9991...
              28118032/28128368            = 0.9996...
             766292835/766512285           = 0.99971...
            1930301618/1930741582          = 0.99977...
            4000783984/4001351168          = 0.99985...
           44303273584/44307987968         = 0.99989...
           85617896265/85625175735         = 0.99991...
          161899964416/161905117184        = 0.99996...
          456280713808/456281771312        = 0.999997...
        52326552030976/52326637800704      = 0.999998...
      1132213192149585/1132213862340015    = 0.9999994...
     27117998061385216/27118008762013184   = 0.99999960...
    226615161368838645/226615243086790155  = 0.99999963...
    286707053950093312/286707152546674688  = 0.99999965...
    692760387785689984/692760539319737216  = 0.99999978...
    794879833279366144/794879862664408064  = 0.99999996...
   5462337996254147968/5462338103823900032 = 0.99999998...

Amiram Eldar, Table of n, a(n) for n = 1..44
Sergei Chernykh, Amicable pairs list.
Shyam Sunder Gupta, Amicable Numbers.

Cf. A063990, A287011.

Cf. A259180 (amicable pairs).

A347770 Conjectured list of numbers which are perfect, amicable, or sociable.

Original entry on oeis.org

6, 28, 220, 284, 496, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 8128, 10744, 10856, 12285, 12496, 14264, 14288, 14316, 14536, 14595, 15472, 17296, 17716, 18416, 19116, 19916, 22744, 22976, 31704, 45946, 47616, 48976, 63020, 66928, 66992, 67095, 69615, 71145, 76084, 79750

Offset: 1

Eric Chen, Sep 13 2021

nonn

By definition, this is the union of A000396, A259180, and A122726. However, at present A122726 is not known to be complete. There is no proof that 564 (for example) is missing from this sequence. - N. J. A. Sloane, Sep 17 2021
Numbers m for which there exists k>=1 such that s^k(m) = m, where s is A001065.
Conjecture: There are no aliquot cycles containing even numbers and odd numbers simultaneously, i.e., every aliquot cycle either has only even numbers or has only odd numbers.

			Known aliquot cycles (sorted by smallest member):
{6}
{28}
{220, 284}
{496}
{1184, 1210}
{2620, 2924}
{5020, 5564}
{6232, 6368}
{8128}
{10744, 10856}
{12285, 14595}
{12496, 14288, 15472, 14536, 14264}
{14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716}
{17296, 18416}
...

David Moews, A list of amicable pairs below 2.01 * 10^11
David Moews, A list of the first 5001 amicable pairs
David Moews, A list of currently known aliquot cycles of length greater than 2 [This list is not known to be complete.]
Jan Munch Pedersen, Tables of Aliquot Cycles (from wayback machine)
Eric Weisstein's World of Mathematics, Perfect number
Eric Weisstein's World of Mathematics, Amicable Pair
Eric Weisstein's World of Mathematics, Sociable Numbers
Wikipedia, Perfect number
Wikipedia, Amicable number
Wikipedia, Sociable number

Cf. A000396, A002025, A002046, A003416, A063990, A122726, A206708, A259180.

Edited with new definition (pointing out that the list is only conjectured to be complete) by N. J. A. Sloane, Sep 17 2021

A385718 Primes p such that there exists prime q < p such that sigma(q+1) = sigma(p+2) = p + q.

Original entry on oeis.org

367, 457, 691, 341647, 909091, 1803421, 2640571, 3076903, 3413191, 5228611, 6152383, 6541477, 6545197, 6695503, 10161133, 10770313, 15319693, 31128511, 31687069, 39946483, 52764031, 58886803, 104494483, 207855001, 283882153, 307912921, 309201751, 529570609, 574061053

Offset: 1

S. I. Dimitrov, Jul 07 2025

nonn

The primes q and p form a P(1, 2)-amicable pair. See Dimitrov link.

			(179, 367) is such a pair because sigma(179+1) = sigma(367+2) = 179 + 367.

S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Cf. A000040, A000203, A063990, A008333, A259180, A385586.

More terms from Jinyuan Wang, Jul 07 2025

A018339 Divisors of 220.

Original entry on oeis.org

1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220

Offset: 1

N. J. A. Sloane

This number has twelve divisors that add up to 504, and 504 - 220 = 284, so 220 is an abundant number. In turn, 284 is a deficient number with six divisors (see A018373) which also add up to 504, and 504 - 284 = 220. - Alonso del Arte, Oct 22 2017

Index entries for sequences related to divisors of numbers

Cf. A063990, A259180.

Divisors[220] (* Alonso del Arte, Oct 22 2017 *)

divisors(220) \\ Charles R Greathouse IV, Jun 22 2017

A018373 Divisors of 284.

Original entry on oeis.org

1, 2, 4, 71, 142, 284

Offset: 1

N. J. A. Sloane

This number has six divisors that add up to 504, and 504 - 284 = 220, so 284 is a deficient number. In turn, 220 is an abundant number with twelve divisors (see A018339) which also add up to 504, and 504 - 220 = 284. - Alonso del Arte, Oct 22 2017

Index entries for sequences related to divisors of numbers

Cf. A063990, A259180.

Divisors[284] (* Alonso del Arte, Oct 22 2017 *)

divisors(284) \\ Charles R Greathouse IV, Jun 22 2017

A180163 Products of pairs of amicable numbers (see A063990).

Original entry on oeis.org

62480, 1432640, 7660880, 27931280, 39685376, 116636864, 179299575, 318523136, 4217802560, 4494828240, 4952759175, 6067699000, 7775676090, 12285798525, 15069863936, 17358731325, 20160203840, 25845386480, 30293400832

Offset: 1

Jonathan Vos Post, Aug 14 2010

For a more reasonable sequence, in which both factors always belong to the same amicable pair, see A180202, which first differs from this sequence at a(9). - Omar E. Pol, Oct 25 2017

			a(1) = 220 * 284 = 62480 = 2^4 * 5 * 11 * 71.
a(2) = 1184 * 1210 = 1432640 = 2^6 * 5 * 11^2 * 37.

Cf. A002025, A002046, A063990, A161005, A180202, A259180.

a(n) = A063990(2*n-1) * A063990(2*n).

A262624 Even amicable numbers.

Original entry on oeis.org

220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 17296, 18416, 63020, 66928, 66992, 76084, 79750, 88730, 122368, 123152, 141664, 142310, 153176, 168730, 171856, 176272, 176336, 180848, 185368, 196724, 202444, 203432, 280540, 308620, 319550, 356408, 365084, 389924, 399592, 430402, 437456, 455344

Offset: 1

Omar E. Pol, Oct 02 2015

nonn

Even numbers that are also amicable numbers.
Intersection of A005843 and A063990.
The first time a pair (x, y) of even amicable numbers ordered by its first element is not adjacent is x = 63020, y = 76084 which correspond to a(15) and a(18), respectively.

Cf. A002025, A002046, A005843, A063990, A259180, A259933, A262625.

t(n)=sigma(n)-n;
is(n)={local(a); a=t(n); a<>n && t(a)==n};
for(n=1, 1e6, if( n%2 == 0 && is(n), print1(n", "))) \\ Altug Alkan, Oct 16 2015

cp's OEIS Frontend

A162884 Half the difference between the larger and smaller term of the n-th amicable pair.

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A275315 Average of amicable pairs (x,y), ordered by the smaller value x given in A002025.

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A275469 Difference between the larger and smaller terms of the n-th amicable pair (x,y) given in A259933.

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A287026 List of pairs of amicable numbers (m, n) with record value of m/n.

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A347770 Conjectured list of numbers which are perfect, amicable, or sociable.

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A385718 Primes p such that there exists prime q < p such that sigma(q+1) = sigma(p+2) = p + q.

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A018339 Divisors of 220.

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A018373 Divisors of 284.

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A180163 Products of pairs of amicable numbers (see A063990).

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