A066539 -id:A066539 - cp's OEIS frontend

A259180 Amicable pairs.

220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 12285, 14595, 17296, 18416, 63020, 76084, 66928, 66992, 67095, 71145, 69615, 87633, 79750, 88730, 100485, 124155, 122265, 139815, 122368, 123152, 141664, 153176, 142310, 168730, 171856, 176336, 176272, 180848, 185368, 203432, 196724, 202444, 280540, 365084

Offset: 1

Omar E. Pol, Jun 20 2015

nonn

A pair of numbers x and y is called amicable if the sum of the proper divisors (or aliquot parts) of either one is equal to the other.
This is A002025 and A002046 interleaved hence the amicable pairs (x < y), ordered by increasing x, are adjacent to each other in the list.
By definition a property of the amicable pair (x, y) is that x + y = sigma(x) = sigma(y).
Amicable numbers A063990 are the terms of this sequence in increasing order.
First differs from A063990 at a(18).
For another version see A259933.
First differs from A259933 at a(17).

			  ------------------------------------
         Amicable pair          Sum
            x      y           x + y
  ------------------------------------
   n    A002025 A002046      A180164
  ------------------------------------
   1       220     284          504
   2      1184    1210         2394
   3      2620    2924         5544
   4      5020    5564        10584
   5      6232    6368        12600
   6     10744   10856        21600
   7     12285   14595        26880
   8     17296   18416        35712
   9     63020   76084       139104
  10     66928   66992       133920
  11     67095   71145       138240
  12     69615   87633       157248
  ...      ...     ...          ...
The sum of the proper divisors (or aliquot parts) of 220 is 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. On the other hand the sum of the proper divisors (or aliquot parts) of 284 is 1 + 2 + 4 + 71 + 142 = 220. Note that 220 + 284 = sigma(220) = sigma(284) = 504. The smallest amicable pair is (220, 284), so a(1) = 220 and a(2) = 284.

Titu Andreescu, Number Theory Trivia: Amicable Numbers
Titu Andreescu, Number Theory Trivia: Amicable Numbers
Anonymous, Amicable Pairs Applet Test
Anonymous, Amicable and Social Numbers [broken link]
S. Chernykh, Amicable Numbers
S. Chernykh, Amicable pairs list
G. D'Abramo, On Amicable Numbers With Different Parity, arXiv:math/0501402 [math.HO], 2005-2007.
E. B. Escott, Amicable numbers, Scripta Mathematica, 12 (1946), 61-72 [Annotated scanned copy]
Leonhard Euler, On amicable numbers, arXiv:math/0409196 [math.HO], 2004-2009.
Mariano Garcia, A Million New Amicable Pairs, J. Integer Sequences, 4 (2001), #01.2.6.
M. García, J. M. Pedersen, H. J. J. te Riele, Amicable pairs, a survey, Report MAS-R0307, Centrum Wiskunde & Informatica.
S. S. Gupta, Amicable Numbers
Hisanori Mishima, Amicable Numbers:first 236 pairs(smaller member<10^8) fully factorized
David Moews, A List Of The First 5001 Amicable Pairs
David and P. C. Moews, A List Of Amicable Pairs Below 2.01*10^11
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
Number Theory List, NMBRTHRY Archives--August 1993
Jan Munch Pedersen, Known Amicable Pairs [Broken link]
Jan Munch Pedersen, Tables of Aliquot Cycles [Broken link]
Ivars Peterson, MathTrek, Appealing Numbers
Ivars Peterson, MathTrek, Amicable Pairs, Divisors and a New Record
Herman J. J. te Riele, On generating new amicable pairs from given amicable pairs, Math. Comp. 42 (1984), 219-223.
Herman J. J. te Riele, Computation of all the amicable pairs below 10^10, Math. Comp., 47 (1986), 361-368 and Supplement pp. S9-S40.
Herman J. J. te Riele, A New Method for Finding Amicable Pairs, Proceedings of Symposia in Applied Mathematics, Volume 48, 1994.
Ed Sandifer, Amicable numbers
Juan Luis Varona, On the Solution of the Equation n = a*k + b*p_k by Means of an Iterative Method, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.5.
Gérard Villemin's Almanach of Numbers, Nombres amiables et sociables
Eric Weisstein's World of Mathematics, Amicable Pair
Wikipedia, Amicable number

Cf. A000203, A001065, A002025, A002046, A063990, A066539, A180164, A180202, A259933.

f[n_] := Block[{s = {}, g, k}, g[x_] := DivisorSigma[1, x] - x; Do[k = g@ i; If[And[g@ k == i, k != i, ! MemberQ[s, i]], s = s~Join~{i, k}], {i, n}]; s]; f@ 300000 (* Michael De Vlieger, Jul 02 2015 *)

A259180_upto(N, L=List(), s)={ forfactored(n=1, N, (s=sigma(n[2]))>2*n[1] && sigma(s-n[1])==s && listput(L, [n[1], s-n[1]]));concat(L)} \\ M. F. Hasler, Oct 11 2019

a(2n-1) = A002025(n); a(2n) = A002046(n).

a(2n-1) + a(2n) = A000203(a(2n-1)) = A000203(a(2n)) = A180164(n).

A180164 The sum of the two numbers in an amicable pair, A002025(n) + A002046(n).

Original entry on oeis.org

504, 2394, 5544, 10584, 12600, 21600, 26880, 35712, 139104, 133920, 138240, 157248, 168480, 224640, 262080, 245520, 294840, 311040, 348192, 357120, 388800, 399168, 645624, 698544, 749952, 756000, 892800, 955206, 1017792, 1048320

Offset: 1

T. D. Noe, Aug 14 2010

nonn

This sequence initially shares many terms with A161005 because small amicable pairs are sometimes consecutive terms in the sorted list of amicable numbers, A063990.
This sequence is sorted by the smaller (abundant) member from A002025, so a(n) is not increasing. - Jeppe Stig Nielsen, Jan 27 2015
Duplicates occur, e.g., a(32)=a(35)=1296000. - Jeppe Stig Nielsen, Jan 27 2015
Comment originally by M. F. Hasler, Dec 14 2013, in A161005: "Also: The common value of sigma(a) = sigma(b) of the amicable pairs (a,b). See A137231 for the analog for amicable triples, and A116148 for quadruples." - Jeppe Stig Nielsen, Jan 27 2015
It is not known if a(n) is always even (see Hagis links). - Jeppe Stig Nielsen, Jan 31 2015
Are all terms abundant (A005101)? The first 10000 terms are. - Ivan N. Ianakiev, Apr 15 2021

			a(9) = A002025(9) + A002046(9) = 63020 + 76084 = 139104.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Peter Hagis, Lower bounds for relatively prime amicable numbers of opposite parity, Math. Comp. 24 (1970), 963-968.
Peter Hagis, Relatively Prime Amicable Numbers of Opposite Parity, Mathematics Magazine, Vol. 43, No. 1 (Jan., 1970), pp. 14-20.
Eric W. Weisstein's World of Mathematics, Pair Sum.

Cf. A002025, A002046, A066539, A259180 (amicable pairs).

s[n_] := DivisorSigma[1,n]-n; smallAmicableQ[n_] := Module[{b=s[n]}, n

Formula

a(n) = A259180(2n-1) + A259180(2n). - Omar E. Pol, Oct 22 2017

A275996 Numbers n whose abundance is 64: sigma(n) - 2n = 64.

Original entry on oeis.org

108, 220, 6808, 8968, 14008, 24448, 66928, 552568, 786208, 1020568, 5303488, 8229568, 10001848, 133685248, 499722448, 2608895488, 4733164768, 7163795488, 13707973408, 14468025568, 16122444736, 27339731968, 34351218688, 34672397728, 35371084288, 69657461248

Offset: 1

Timothy L. Tiffin, Aug 16 2016

Any term x = a(m) of this sequence can be used with any term y of A275997 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable.
The smallest amicable pair is (220, 284) = (a(2), A275997(2)) = (A063990(1), A063990(2)), where 284 - 220 = 64 is the abundance of 220 and the deficiency of 284.
The amicable pair (66928, 66992) = (a(7), A275997(11)) = (A063990(18), A063990(19)), and 66992 - 66928 = 64 is the abundance of 66928 and the deficiency of 66992.

			a(1) = 108, since sigma(108) - 2*108 = 280 - 216 = 64.

Max Alekseyev, Table of n, a(n) for n = 1..63

Subsequence of A005101.

Cf. A002025, A063990, A275997, A088831, A088832, A088833, A141547, A175989, A275701, A066539, A259180.

isok(n) = sigma(n) - 2*n == 64; \\ Michel Marcus, Dec 30 2016

a(14)-a(15) from Michel Marcus, Dec 30 2016
a(16)-a(21) from Lars Blomberg, Jan 12 2017
Terms a(22) onward from Max Alekseyev, Aug 27 2025

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

Original entry on oeis.org

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).

A275470 Half the difference between the larger and smaller terms of the n-th amicable pair (x,y) given in A259933.

Original entry on oeis.org

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

Offset: 1

Timothy L. Tiffin, Jul 28 2016

nonn

Each term represents the radius of an interval (x,y), where x (A260086) and y (A260087) form a pair of amicable numbers (A259933).  The midpoint and length of each interval can be found in A275316 and A275469, respectively.
A term will be odd if and only if y-x = 2 mod 4.  This occurs when x and y have the same parity but their average has the opposite parity.
This sequence is a rearrangement of A162884 (which is based on A002025, A002046, and A066539).  The first ten indices for which a(n) does not equal A162884(n) are n = 9, 10, 11, 15, 16, 33, 34, 35, 41, 42.

			a(1) = (284-220)/2 = 64/2 = 32, a(2) = (1210-1184)/2 = 26/2 = 13, and a(3) = (2924-2620)/2 = 304/2 = 152.

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, A260086, A260087, A162884, A259933, A275316, A275469.

a(n) = [A260087(n) - A260086(n)]/2 = A275469(n)/2.

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

A178542 Difference A063990(2n)-A063990(2n-1) between amicable numbers.

Original entry on oeis.org

64, 26, 304, 544, 136, 112, 2310, 1120, 3908, 103, 1530, 3666, 1097, 21780, 784, 15660, 646, 15554, 4416, 4512, 11356, 988, 28080, 36858, 24840, 30810, 17888, 17150, 11680, 3510, 9536, 11440, 9328, 1724, 5184, 13888, 13140, 61110, 18504

Offset: 1

Roger L. Bagula, May 29 2010

nonn

Note that these are not differences between two members of amicable pairs

(see A066539) because A063990 is sorted and may separate pairs of amicable numbers.

Cf. A063990.

(*A063990 Amicable numbers.*);
s[n_] := DivisorSigma[1, n] - n;
AmicableNumberQ[n_] := If[Nest[s, n, 2] == n && ! s[n] == n, True, False];
a = Select[Range[10^6], AmicableNumberQ[ # ] &];
Table[a[[n + 1]] - a[[n]], {n, 1, Length[a], 2}]

Formula clarified by the Assoc. Eds. of the OEIS - Jun 07 2010

A306613 First differences of A063990 (amicable numbers arranged in increasing order).

Original entry on oeis.org

64, 900, 26, 1410, 304, 2096, 544, 668, 136, 4376, 112, 1429, 2310, 2701, 1120, 44604, 3908, 64, 103, 2520, 1530, 4939, 3666, 7883, 1097, 11755, 21780, 103, 784, 1003, 15660, 1849, 646, 10866, 15554, 3126, 4416, 64, 4512, 4520, 11356, 5720, 988, 77108, 28080, 10930

Offset: 1

Conor Coons, Feb 28 2019

nonn

a(n) is the difference between the n-th and (n+1)-th amicable numbers when ordered by increasing value.

For 1 <= k <= 8, a(2k-1) is the difference between the larger and the smaller terms of the k-th amicable pair, and for 1 <= k <= 8, a(2k) is the difference between the smaller term of the (k+1)-th pair and the larger term of the k-th pair. Beginning with the 9th pair (63020,76084), the pairs ordered by their first element are no longer adjacent. - Bernard Schott, Mar 09 2019

			a(2) = amicable(3) - amicable(2) = 1184 - 284 = 900.
From _Bernard Schott_, Mar 10 2019: (Start)
a(1) = 284 - 220 = 64 is the difference between the larger and the smaller terms of the first amicable pair.
a(4) = 2620 - 1210 = 1410 is the difference between the smaller term of the third amicable pair and the larger term of the second amicable pair. (End)

Cf. A063990 (amicable numbers), A306612.
Cf. A002025, A002046.
Cf. A066539 (difference between larger and smaller terms of n-th amicable pair).
Cf. A139228 (first differences of perfect numbers).

clear
clc
A = zeros(100000,1);
parfor n = 1:1:100000
     f = find(rem(n, 1:floor(sqrt(n))) == 0);
     f = unique([1, n, f, fix(n./f)]);
     A(n) = sum(f) - n;
end
D = [];
d = 1;
for a = 1:1:100000
    for b = 1:1:100000
        if A(a) == b && A(b) == a && a~=b
            D(d) = a;
            d = d+1;
        end
    end
end
D
difference = diff(D)

a(n) = A063990(n+1) - A063990(n). - Michel Marcus, Apr 08 2019

More terms from Michel Marcus, Mar 09 2019

cp's OEIS Frontend

A259180 Amicable pairs.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A180164 The sum of the two numbers in an amicable pair, A002025(n) + A002046(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A275996 Numbers n whose abundance is 64: sigma(n) - 2n = 64.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A275470 Half the difference between the larger and smaller terms of the n-th amicable pair (x,y) given in A259933.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Examples

References