A180164 -id:A180164 - cp's OEIS frontend

A063990 Amicable numbers.

220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 12285, 14595, 17296, 18416, 63020, 66928, 66992, 67095, 69615, 71145, 76084, 79750, 87633, 88730, 100485, 122265, 122368, 123152, 124155, 139815, 141664, 142310

Offset: 1

N. J. A. Sloane, Sep 18 2001

nonn

A pair of numbers x and y is called amicable if the sum of the proper divisors of either one is equal to the other. The smallest pair is x = 220, y = 284.
The sequence lists the amicable numbers in increasing order. Note that the pairs x, y are not adjacent to each other in the list. See also A002025 for the x's, A002046 for the y's.
Theorem: If the three numbers p = 3*(2^(n-1)) - 1, q = 3*(2^n) - 1 and r = 9*(2^(2n-1)) - 1 are all prime where n >= 2, then p*q*(2^n) and r*(2^n) are amicable numbers. This 9th century theorem is due to Thabit ibn Kurrah (see for example, the History of Mathematics by David M. Burton, 6th ed., p. 510). - Mohammad K. Azarian, May 19 2008
The first time a pair ordered by its first element is not adjacent is x = 63020, y = 76084 which correspond to a(17) and a(23), respectively. - Omar E. Pol, Jun 22 2015
For amicable pairs see A259180 and also A259933. First differs from A259180 (amicable pairs) at a(18). - Omar E. Pol, Jun 01 2017
Sierpiński (1964), page 176, mentions Erdős's work on the number of pairs of amicable numbers <= x. - N. J. A. Sloane, Dec 27 2017
Kanold (1954) proved that the asymptotic upper density of amicable numbers is < 0.204 and Erdős (1955) proved that it is 0. - Amiram Eldar, Feb 13 2021

Scott T. Cohen, Mathematical Buds, Ed. Harry D. Ruderman, Vol. 1, Chap. VIII, pp. 103-126, Mu Alpha Theta, 1984.
Clifford A. Pickover, The Math Book, Sterling, NY, 2009; see p. 90.
Wacław Sierpiński, Elementary Theory of Numbers, Panst. Wyd. Nauk, Warsaw, 1964.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 137-141.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 145-147.

T. D. Noe, Table of n, a(n) for n = 1..77977 (terms < 10^14 from Pedersen's tables)
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]
Jonathan Bayless and Dominic Klyve, On the sum of reciprocals of amicable numbers, Integers, Vol. 11A (2011), Article 5.
Sergei Chernykh, Table of n, a(n) for n = 1..823818, zipped file (results of an exhaustive search for all amicable pairs with smaller member < 10^17).
Sergei Chernykh, Amicable pairs list.
Germano D'Abramo, On Amicable Numbers With Different Parity, arXiv:math/0501402 [math.HO], 2005-2007.
Paul Erdős, On amicable numbers, Pub. Math. Debrecen, Vol. 4 (1955), pp. 108-111.
Leonhard Euler, On amicable numbers, arXiv:math/0409196 [math.HO], 2004-2009.
Steven Finch, Amicable Pairs and Aliquot Sequences, 2013. [Cached copy, with permission of the author]
Mariano García, A Million New Amicable Pairs, J. Integer Sequences, Vol. 4 (2001), Article #01.2.6.
Mariano García, Jan Munch Pedersen, Herman te Riele, Amicable pairs, a survey, Report MAS-R0307, Centrum Wiskunde & Informatica.
Hans-Joachim Kanold, Über die Dichten der Mengen der vollkommenen und der befreundeten Zahlen, Mathematische Zeitschrift, Vol. 61 (1954), pp. 180-185.
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
Hanh My Nguyen and Carl Pomerance, The reciprocal sum of the amicable numbers, Mathematics of Computation, Vol. 88, No. 317 (2019), pp. 1503-1526, alternative link.
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.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
Ivars Peterson, Appealing Numbers, MathTrek, 2001.
Ivars Peterson, Amicable Pairs, Divisors and a New Record, MathTrek, 2004.
Paul Pollack, Quasi-Amicable Numbers are Rare, J. Int. Seq., Vol. 14 (2011), Article # 11.5.2.
Carl Pomerance, On amicable numbers, in: C. Pomerance and M. Rassias M. (eds.), Analytic number theory, Springer, Cham, 2015, pp. 321-327; alternative link.
Herman J. J. te Riele, On generating new amicable pairs from given amicable pairs, Math. Comp., Vol. 42, No. 165 (1984), pp. 219-223.
Herman J. J. te Riele, Computation of all the amicable pairs below 10^10, Math. Comp., Vol. 47, No. 175 (1986), pp. 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.
Gérard Villemin's Almanach of Numbers, Nombres amiables et sociables.
Eric Weisstein's World of Mathematics, Amicable Pair.
Wikipedia, Amicable number.

Union of A002025 and A002046.
A180164 (gives for each pair (x, y) the value x+y = sigma(x)+sigma(y)).
Cf. A259180.

F:= proc(t) option remember; numtheory:-sigma(t)-t end proc:
select(t -> F(t) <> t and F(F(t))=t, [$1.. 200000]); # Robert Israel, Jun 22 2015

s[n_] := DivisorSigma[1, n] - n; AmicableNumberQ[n_] := If[Nest[s, n, 2] == n && ! s[n] == n, True, False]; Select[Range[10^6], AmicableNumberQ[ # ] &] (* Ant King, Jan 02 2007 *)
Select[Tally[Sort/@Table[{n,DivisorSigma[1,n]-n},{n,200000}]],#[[2]]==2&][[;;,1]]//Flatten//Sort (* Harvey P. Dale, Jan 13 2025 *)

aliquot(n)=sigma(n)-n
isA063990(n)={if (n>1, local(a);a=aliquot(n);a<>n && aliquot(a)==n)} \\ Michael B. Porter, Apr 13 2010

from sympy import divisors
A063990 = [n for n in range(1,10**5) if sum(divisors(n))-2*n and not sum(divisors(sum(divisors(n))-n))-sum(divisors(n))] # Chai Wah Wu, Aug 14 2014

Pomerance shows that there are at most x/exp(sqrt(log x log log log x)/(2 + o(1))) terms up to x for sufficiently large x. - Charles R Greathouse IV, Jul 21 2015

Sum_{n>=1} 1/a(n) is in the interval (0.0119841556, 215) (Nguyen and Pomerance, 2019; an upper bound 6.56*10^8 was given by Bayless and Klyve, 2011). - Amiram Eldar, Oct 15 2020

A002025 Smaller of an amicable pair: (a,b) such that sigma(a) = sigma(b) = a+b, a < b.

Original entry on oeis.org

220, 1184, 2620, 5020, 6232, 10744, 12285, 17296, 63020, 66928, 67095, 69615, 79750, 100485, 122265, 122368, 141664, 142310, 171856, 176272, 185368, 196724, 280540, 308620, 319550, 356408, 437456, 469028, 503056, 522405, 600392, 609928

Offset: 1

N. J. A. Sloane

Sometimes called friendly numbers, but this usage is deprecated.
All terms are abundant (A005101). - Michel Marcus, Mar 10 2013
See A125490-A125492 and A137231 for amicable triples, A036471-A036474 and A116148 for amicable quadruples, and A233553 for amicable quintuples. - M. F. Hasler, Dec 14 2013
This sequence is strictly increasing (and A002046, which contains the larger (deficient) number in each pair, is sorted by this sequence). - Jeppe Stig Nielsen, Jan 27 2015
For the related amicable pairs see A259180. - Omar E. Pol, Jul 15 2015
Pomerance (1981) shows that there are at most x*exp(-log(x)^(1/3)) terms of this sequence up to x. In particular, as originally demonstrated by Erdős, this sequence has density 0. - Charles R Greathouse IV, Aug 17 2017

Mariano Garcia, Jan Munch Pedersen and Herman te Riele, Amicable pairs - a survey, pp. 179-196 in: Alf van der Poorten and Andres Stein (eds.), High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Fields Institute Communications, AMS, Providence RI, 2004.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 48-49.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Sergei Chernykh, Table of n, a(n) for n = 1..415523 [All terms up to 10^17. Terms 39375 through 415523 were computed by Sergei Chernykh]
J. Alanen, O. Ore and J. Stemple, Systematic computations on amicable numbers, Math. Comp., 21 (1967), 242-245.
J. Bell, A translation of Leonhard Euler's..., arXiv:math/0409196 [math.HO], 2004-2009.
W. Borho and H. Hoffmann, Breeding Amicable Numbers in Abundance, Math. Comp., 46 (1986), 281-293.
S. Chernykh, Amicable pairs list.
Paul Erdős, On amicable numbers, Publ. Math. Debrecen 4 (1955), pp. 108-111.
E. B. Escott, Amicable numbers, Scripta Mathematica, 12 (1946), 61-72. [Annotated scanned copy]
L. Euler, De numeris amicabilibus, Opuscula varii argumetii, pages 23-107, 1750. Reprinted in Opera mathematica: Series prima. Volumen II, Leonhardi Euleri commentationes arithmeticae. Sub ausp. soc. scient. nat. Helv., Teubner, Leipzig, Series I, Vol. 1915, pp. 86-162.
M. Garcia, A Million New Amicable Pairs, J. Integer Seqs., Vol. 4 (2001), #01.2.6.
Mariano García, Jan Munch Pedersen, and Herman J. J. te Riele, Amicable pairs, a survey, Report MAS-R0307, 2003, Centrum Wiskunde en Informatica.
Mariano García, Jan Munch Pedersen, and Herman J. J. te Riele, Amicable pairs, a survey, Fields Institute Comm. (2004) Vol. 41.
S. S. Gupta, Amicable Numbers.
E. J. Lee, Amicable Numbers and the Bilinear Diophantine Equation, Math. Comp., 22 (1968), 181-187.
Hisanori Mishima, First 236 amicable pairs.
D. Moews, Perfect, amicable and sociable numbers.
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
J. O. M. Pedersen, Known Amicable Pairs. [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
Carl Pomerance, On the distribution of amicable numbers, J. reine angew. Math. 293/294 (1977), pp. 217-222.
Carl Pomerance, On the distribution of amicable numbers, II, J. reine angew. Math. 325 (1981), pp. 183-188.
H. J. J. te Riele, Four large amicable pairs, Math. Comp., 28 (1974), 309-312.
H. J. J. te Riele, Computation of all the amicable pairs below 10^10, Math. Comp., 47 (1986), 361-368 and Supplement pp. S9-S40.
H. J. J. te Riele et al., Table of Amicable Pairs between 10^10 and 10^52, Note NM-N8603, Department of Numerical Mathematics, Centre for Mathematics and Computer Science, Amsterdam, 1986. (Warning: file size is 65MB.)
T. Trotter, Jr., Amicable Numbers, archived from the original.
Eric Weisstein's World of Mathematics, Amicable Pair.

Cf. A000203, A002046, A063990, A066873, A180164, A259180.

Reap[For[n = 1, n <= 10^6, n++, If[(s = DivisorSigma[1, n]) > 2n && DivisorSigma[1, s - n] == s, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 09 2015, after M. F. Hasler *)

aliquot(n)=sigma(n)-n
isA002025(n)={if (n>1, local(a);a=aliquot(n);a>n && aliquot(a)==n)} \\ Michael B. Porter, Apr 11 2010

for(n=1,1e6,(s=sigma(n))>2*n && sigma(s-n)==s && print1(n",")) \\ M. F. Hasler, Dec 14 2013

forfactored(n=1,10^6, t=sigma(n[2])-n[1]; if(t>n[1] && sigma(t)==n[1]+t, print1(n[1]", "))) \\ Charles R Greathouse IV, Aug 17 2017

a(n) = A259180(2n-1) = A180164(n) - A259180(2n) = A180164(n) - A002046(n). - Omar E. Pol, Jul 15 2015

More terms from Larry Reeves (larryr(AT)acm.org), Oct 24 2000

A002046 Larger of amicable pair.

Original entry on oeis.org

284, 1210, 2924, 5564, 6368, 10856, 14595, 18416, 76084, 66992, 71145, 87633, 88730, 124155, 139815, 123152, 153176, 168730, 176336, 180848, 203432, 202444, 365084, 389924, 430402, 399592, 455344, 486178, 514736, 525915, 669688, 686072

Offset: 1

N. J. A. Sloane

The elements 76084, 123152, etc. are intentionally out of numerical order so that a(n) and A002025(n) form amicable pairs. - Michael B. Porter, Apr 17 2010
All terms are deficient (A005100). - Michel Marcus, Mar 10 2013
For the related amicable pairs see A259180. - Omar E. Pol, Jul 15 2015

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
For additional references see A002025.

T. D. Noe and Sergei Chernykh, Table of n, a(n) for n = 1..415523 [All terms such that the smaller number A002025(n) is < 10^17. Terms 39375 through 415523 were computed by Sergei Chernykh]
J. Bell, A translation of Leonhard Euler's "On amicable numbers", arXiv:math/0409196 [math.HO], 2004-2009.
S. Chernykh, Amicable pairs list
E. B. Escott, Amicable numbers, Scripta Mathematica, 12 (1946), 61-72 [Annotated scanned copy]
M. Garcia, A Million New Amicable Pairs, J. Integer Seqs., Vol. 4 (2001), #01.2.6.
S. S. Gupta, Amicable Numbers
Hisanori Mishima, First 236 amicable pairs
David Moews, Perfect, amicable and sociable numbers
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
J. O. M. Pedersen, Known Amicable Pairs [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
T. Trotter, Jr., Amicable Numbers
Eric Weisstein's World of Mathematics, Amicable Pair

Cf. A002025, A063990, A180164, A259180.

f:= proc(t) uses numtheory; local s;
  s:= sigma(t) - t; s > t and sigma(s) - s = t
end proc;
Am1:= select(f,[$1..10^6]);
map(numtheory:-sigma,Am1); # Robert Israel, Jul 16 2015

amicableQ[n_] := With[{s = DivisorSigma[1, n] - n}, r = n != s && n == DivisorSigma[1, s] - s; If[r, mate[n] = s; True, False]]; mate /@ Select[ Range[lim], amicableQ[#] && # < mate[#] &] (* Jean-François Alcover, Sep 20 2011 *)
Table[DivisorSigma[1, A002025[n]] - A002025[n], {n, 50}] (* T. D. Noe, Sep 20 2011 *)

aliquot(n)=sigma(n)-n
isA002046(n)={if (n>1, local(a);a=aliquot(n);aMichael B. Porter, Apr 17 2010

a(n) = A259180(2n) = A180164(n) - A259180(2n-1) = A180164(n) - A002025(n). - Omar E. Pol, Jul 15 2015

More terms from Larry Reeves (larryr(AT)acm.org), Oct 25 2000

A259180 Amicable pairs.

Original entry on oeis.org

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).

A161005 Sums of adjacent amicable numbers, a(n) = A063990(2n-1) + A063990(2n).

Original entry on oeis.org

504, 2394, 5544, 10584, 12600, 21600, 26880, 35712, 129948, 134087, 140760, 155834, 176363, 222750, 245520, 263970, 283974, 321906, 348128, 357184, 382092, 405876, 589160, 675958, 755008, 829994, 892800, 955206, 1017792, 1048320

Offset: 1

Claudio Meller, Jun 01 2009

Warning: The numbers being summed will not always belong to the same amicable pair. See A180164 for the sums of amicable pairs. - Jeppe Stig Nielsen, Jan 27 2015

			a(1) = 504 = 220 + 284.
a(2) = 2394 = 1184 + 1210.
a(3) = 5544 = 2620 + 2694.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A063990, A259180.

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}]
(* Roger L. Bagula, May 29 2010, based on Ant King's Mathematica program from A063990 *)

Corrected and extended by Roger L. Bagula, May 29 2010

Edited by N. J. A. Sloane, Aug 14 2010, at the suggestion of Jason G. Wurtzel

A201915 Each row of triangle T(n,k) has a sorted list of n values such that sigma(T(n,k)) = A007368(n).

Original entry on oeis.org

1, 6, 11, 14, 15, 23, 42, 62, 69, 77, 30, 46, 51, 55, 71, 60, 78, 92, 123, 143, 167, 114, 135, 158, 177, 203, 209, 239, 132, 140, 182, 188, 195, 249, 287, 299, 120, 174, 184, 190, 267, 295, 319, 323, 359, 204, 220, 224, 246, 284, 286, 334, 415, 451, 503

Offset: 1

T. D. Noe, Jan 24 2012

The first and last terms of each row are given in A184393 and A184394.

Note that the integers in the 10th row have sigma(n)=504 (A180164(1)) and thus include A002025(1) and A002046(1). - Michel Marcus, Oct 22 2013

			Triangle:
1
6,    11
14,   15,  23
42,   62,  69,  77
30,   46,  51,  55,  71
60,   78,  92, 123, 143, 167
114, 135, 158, 177, 203, 209, 239
132, 140, 182, 188, 195, 249, 287, 299
120, 174, 184, 190, 267, 295, 319, 323, 359
204, 220, 224, 246, 284, 286, 334, 415, 451, 503

T. D. Noe, Rows n = 1..100 of triangle, flattened

A328009 Irregular array read by rows in which row n lists the divisors of the n-th term of the sequence of amicable pairs (A259180).

Original entry on oeis.org

1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220, 1, 2, 4, 71, 142, 284, 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592, 1184, 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605, 1210, 1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310, 2620, 1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462, 2924, 1, 2, 4, 5, 10, 20, 251, 502, 1004, 1255

Offset: 1

Omar E. Pol, Oct 01 2019

Row sums give a sequence formed by the terms of A180164 repeated as follows: 504, 504, 2394, 2394, 5544, 5544, ...

			Array begins:
1, 2, 4,  5,  10,  11,  20,  22,   44,  55,   110,  220;
1, 2, 4, 71, 142, 284;
1, 2, 4,  8,  16,  32,  37,  74,  148,  296,  592, 1184;
1, 2, 5, 10,  11,  22,  55, 110,  121,  242,  605, 1210;
1, 2, 4,  5,  10,  20, 131, 262,  524,  655, 1310, 2620;
1, 2, 4, 17,  34,  43,  68,  86,  172,  731, 1462, 2924;
1, 2, 4,  5,  10,  20, 251, 502, 1004, 1255, 2510, 5020;
1, 2, 4, 13,  26,  52, 107, 214,  428, 1391, 2782, 5564,
1, 2, 4,  8,  19,  38,  41,  76,   82,  152,  164,  328, 779, 1558, 3116, 6232;
1, 2, 4,  8,  16,  32, 199, 398,  796, 1592, 3184, 6368;
...

Right border gives A259180 (amicable pairs).
The length of row n is A328043(n).
Column 1 gives A000012.
Cf. A002025, A002046, A018339, A018373, A027750, A180164, A328063, A328064,  A328065.

With[{s = Array[{#, DivisorSigma[1, #] - #} &, 6000]}, Flatten@ Divisors@ DeleteDuplicates[Sort /@ Select[Reverse /@ s, And[! FreeQ[s, #], UnsameQ @@ #] &]]] (* Michael De Vlieger, Oct 08 2019 *)

A259953 The sum (in nondecreasing order) of the two numbers in an amicable pair.

Original entry on oeis.org

504, 2394, 5544, 10584, 12600, 21600, 26880, 35712, 133920, 138240, 139104, 157248, 168480, 224640, 245520, 262080, 294840, 311040, 348192, 357120, 388800, 399168, 645624, 698544, 749952, 756000, 892800, 955206, 1017792, 1048320, 1270080, 1296000, 1296000, 1315440, 1347840, 1451520, 1522800, 1666560, 1781136, 1879200, 2041200

Offset: 1

Omar E. Pol, Jul 10 2015

nonn

Also the common value of sigma(x) = sigma(y) of the amicable pairs (x < y) ordered by nondecreasing sum (x + y). See A259933.
Duplicates occur, e.g., a(32) = a(33) = 1296000.
Another version of A180164.
First differs from both A161005 and A180164 at a(9).

			------------------------------------------
      A m i c a b l e   p a i r      Sum
------------------------------------------
n     A260086(n)  +  A260087(n)  =   a(n)
------------------------------------------
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       66928          66992       133920
10      67095          71145       138240
11      63020          76084       139104
12      69615          87633       157248
...       ...            ...          ...
32     609928         686072      1296000
33     643336         652664      1296000
...

Cf. A000203, A063990, A161005, A180164, A259180, A259933, A260086, A260087.

a(n) = A259933(2n-1) + A259933(2n) = A260086(n) + A260087(n).

A291422 List of pairs of amicable numbers (m,n) where the sum of the pair is divisible by 10.

Original entry on oeis.org

6232, 6368, 10744, 10856, 12285, 14595, 66928, 66992, 67095, 71145, 79750, 88730, 100485, 124155, 122265, 139815, 122368, 123152, 141664, 153176, 142310, 168730, 176272, 180848, 185368, 203432, 356408, 399592, 437456, 455344, 522405, 525915, 600392, 669688, 609928, 686072

Offset: 1

Zoltan Galantai, Aug 22 2017

The sequence lists those amicable pairs (m,n) in increasing order where the sum of the amicable pair is divisible by ten.
Up to the first 5001 amicable pairs, 88.1% of the sums satisfy this condition (up to the first 100 amicable pairs: 74%; up to the first 1000: 82.5%; up to 2000: 85.25%).  So the conjecture here is that as the number of the amicable numbers approaches infinity, the percentage of the sums of the amicable pairs divisible by ten approaches 100%. [corrected by Paul Zimmermann, Feb 05-06 2019]
Among the 1947667 pairs up to 19 digits from Sergei Chernykh's database, there are 1872573 pairs with m+n divisible by ten, thus about 96.14%. - Paul Zimmermann, Feb 07 2019

			The sum of 6232 and 6368 is divisible by ten, thus the (6232, 6368) amicable pair belongs to the sequence. On the other hand, the (220, 284) amicable pair does not qualify since its sum is 504.

Richard K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1994, pp. 55-58.
Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chappman and HALL/CRC, 2003, pp. 67-69.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sergei Chernykh, Amicable pairs list.
Zoltan Galantai, List of the amicable pairs where the sum divisible by ten smaller and larger amicable numbers; sums (the first 4406 pairs).
Zoltan Galantai, List of the first 5001 amicable pairs with their sums denoting whether the sum is divisible by ten or not.
Eric Weisstein's World of Mathematics, Amicable Pair.
Eric Weisstein's World of Mathematics, Sociable Numbers.

Cf. A002025, A002046, A063990, A180164, A259180, A259933, A291550.

lista(nn) = {for (n=1, nn, spd = sigma(n)-n; if ((spd > n) && (sigma(spd)-spd == n) && !((n + spd) % 10), print1(n, ", ", spd, ", ")););} \\ Michel Marcus, Aug 26 2017

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.

cp's OEIS Frontend

A063990 Amicable numbers.

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A002025 Smaller of an amicable pair: (a,b) such that sigma(a) = sigma(b) = a+b, a < b.

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A002046 Larger of amicable pair.

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A259180 Amicable pairs.

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A161005 Sums of adjacent amicable numbers, a(n) = A063990(2n-1) + A063990(2n).

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A201915 Each row of triangle T(n,k) has a sorted list of n values such that sigma(T(n,k)) = A007368(n).

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A328009 Irregular array read by rows in which row n lists the divisors of the n-th term of the sequence of amicable pairs (A259180).

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