A259180 -id:A259180 - cp's OEIS frontend

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A275316 Average of amicable pairs (x,y), ordered by the sum x+y given in A259953.

252, 1197, 2772, 5292, 6300, 10800, 13440, 17856, 66960, 69120, 69552, 78624, 84240, 112320, 122760, 131040, 147420, 155520, 174096, 178560, 194400, 199584, 322812, 349272, 374976, 378000, 446400, 477603, 508896, 524160, 635040, 648000, 648000, 657720, 673920, 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 (A260086) and y (A260087) form a pair of amicable numbers (A259933).  The length and radius of each interval can be found in A275469 and A275470, respectively.
This sequence is monotonic (specifically, nondecreasing), since x+y (A259953) is nondecreasing.  For a nonmonotonic ordering of these averages, see A275315.
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) = (   66928 +    66992)/2 =   133920/2 =    66960.
a( 10) = (   67095 +    71145)/2 =   138240/2 =    69120.
a( 11) = (   63020 +    76084)/2 =   139104/2 =    69552.
...      ...                 ...          ...         ...
a( 15) = (  122368 +   123152)/2 =   245520/2 =   122760.
a( 16) = (  122265 +   139815)/2 =   262080/2 =   131040.
a( 17) = (  141664 +   153176)/2 =   294840/2 =   147420.
...      ...                 ...          ...         ...
a( 32) = (  609928 +   686072)/2 =  1296000/2 =   648000.
a( 33) = (  643336 +   652664)/2 =  1296000/2 =   648000.
...      ...                 ...          ...         ...
a(107) = ( 9478910 + 11049730)/2 = 20528640/2 = 10264320.
a(108) = (10254970 + 10273670)/2 = 20528640/2 = 10264320.
...      ...                 ...          ...         ...
a(139) = (17754165 + 19985355)/2 = 37739520/2 = 18869760.
a(140) = (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. A260086, A260087, A259180, A259933, A259953, A275315, A275469, A275470.

With[{s = PositionIndex@ Array[DivisorSigma[1, #] &, 10^6]}, Flatten@ Map[Mean, Apply[Join, Map[Function[n, Select[Subsets[Lookup[s, n], {2}], Total@ # == n &]], Sort@ Select[Keys@ s, Length@ Lookup[s, #] > 1 &]]]]] (* Michael De Vlieger, Oct 22 2017 *)

a(n) = [A260086(n) + A260087(n)]/2 = A259953(n)/2.

A383239 Integers k such that there exists an integer 0

Original entry on oeis.org

1740, 7776, 22428, 55968, 106140, 143910, 198792, 246510, 309582, 326196, 411138, 421596, 428256, 590112, 639288, 697158, 870552, 941094, 958716, 1060956, 1087776, 1105884, 1269828, 1341660, 1361568, 1447620, 1495494, 1512810, 1626324, 1727940, 1819392

Offset: 1

S. I. Dimitrov, Apr 20 2025

S. I. Dimitrov introduced the notion of (alpha_1,...,alpha_k)-multiamicable k-tuples.

The asymptotic density of (alpha_1, alpha_2)-multiamicable pairs relative to the positive integers is 0.

			For k=2, alpha_1=1, alpha_2=2 we have (1560, 1740), (7380, 7776), (20664, 22428), (543456, 590112), (588744, 639288),

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

Subsequence of A259302.

Cf. A000203, A005820, A063990, A259180, A125490, A036474, A259303, A292365.

isok(k) = my(s=sigma(k), m=s-2*k); m>0 && mMichel Marcus, Apr 28 2025

We say that the natural numbers n_1,..., n_k form an (alpha_1,...,alpha_k)-multiamicable k-tuple if sigma(n_1)=sigma(n_2)=...=sigma(n_k)=alpha_1n_1+alpha_2n_2+...+alpha_kn_k, where alpha_1,...,alpha_k are positive integers, where sigma(n) is the sum of the divisors of n.

More terms from Sean A. Irvine, May 04 2025

A180202 The product of the two numbers in an amicable pair, A002025(n) * A002046(n).

Original entry on oeis.org

62480, 1432640, 7660880, 27931280, 39685376, 116636864, 179299575, 318523136, 4794813680, 4483640576, 4773473775, 6100571295, 7076217500, 12475715175, 17094480975, 15069863936, 21699524864, 24011966300, 30304399616

Offset: 1

T. D. Noe, Aug 15 2010

nonn

This sequence initially shares many terms with A180163 because small amicable pairs are sometimes consecutive terms in the sorted list of amicable numbers, A063990.

First differs from A180163 at a(9). - Omar E. Pol, Oct 25 2017

			a(9) = A002025(9) * A002046(9) = 63020 * 76084 = 4794813680.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A002025, A002046, A180163, A259180.

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

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

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Author

Omar E. Pol, Jul 10 2015

Keywords

nonn

Comments

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).

Examples

------------------------------------------ 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 ...

Crossrefs

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

Formula

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

A262622 Amicable pairs of even numbers.

Original entry on oeis.org

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

Offset: 1

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Author

Omar E. Pol, Nov 09 2015

Keywords

nonn
tabf

Comments

If there are no amicable pairs whose members have distinct parity then this is also the even terms of A259180.

First differs from A063990, A259180, A259933 at a(13).

First differs from A262624 at a(16).

Links

Amiram Eldar, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A002025, A002046, A005843, A063990, A259180, A259933, A262623, A262624.

Programs

PARI

listap(nn) = {forstep(n=2, nn, 2, m = sigma(n)-n; if ((m > n) && (n==sigma(m)-m), print1(n, ", ", m, ", ")););} \\ Michel Marcus, Nov 14 2015

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

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Author

Zoltan Galantai, Aug 22 2017

Keywords

nonn
tabf

Comments

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

Examples

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.

References

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.

Links

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.

Crossrefs

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

Programs

PARI

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

A307962 Lesser of coreful amicable numbers pair: numbers (m, n) such that csigma(m) = csigma(n) = m + n, where csigma(n) is the sum of the coreful divisors of n (A057723).

Original entry on oeis.org

1718200, 4818880, 5154600, 12027400, 14456640, 22336600, 29209400, 32645800, 33732160, 36082200, 39518600, 49827800, 53264200, 62645440, 63573400, 67009800, 70446200, 73882600, 80755400, 81920960, 87628200, 91064600, 91558720, 97937400, 101196480, 101373800

Offset: 1

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Author

Amiram Eldar, May 08 2019

Keywords

nonn

Comments

The larger counterparts are in A307963.

If (m, n) is an amicable pair (A259180), then the pair (m*k, n*k) with k=rad(m*n) is a coreful amicable pair (rad(i)=A007947(i) is the squarefree kernel of i), and so are all the pairs (m*k*s, n*k*s) where s is a squarefree number with gcd(s, k) = 1. Proof: k = rad(m*n) = rad(m)*rad(n)/rad(gcd(m,n)), csigma(m*k) = csigma(m*rad(m)*j) where j = rad(n)/rad(gcd(m,n)) is squarefree and coprime to m*rad(m), so csigma(m*k) = j * csigma(m*rad(m)) = j * rad(m)* sigma(m) = rad(m)*rad(n)/rad(gcd(m,n)) * sigma(m) = rad(m)*rad(n)/rad(gcd(m,n)) * (n+m) = k *(n+m) = csigma(n*k).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007947, A057723, A063990, A259180, A307958, A307959, A307963.

Programs

Mathematica

f[p_,e_] := (p^(e+1)-1)/(p-1)-1; csigma[1]=1; csigma[n_] := Times @@ (f @@@ FactorInteger[n]); s={}; Do[m = csigma[n] - n; If[m > n && csigma[m] - m == n, AppendTo[s, n]], {n, 1, 10^8}]; s

Extensions

Wrong terms corrected by Amiram Eldar, Dec 02 2019

A383483 Numbers k such that k = sigma(m)-m where m = sigma(3*k)-3*k.

Original entry on oeis.org

3, 15, 5919, 118719, 179871, 33750303

Offset: 1

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Author

S. I. Dimitrov, Apr 28 2025

Keywords

nonn
more

Comments

S. I. Dimitrov introduced the notion of (alpha, beta)-amicable pairs.

Examples

For alpha=1, beta=3 we have (3, 4), (15, 33), (5919, 7905). Here (3, 4) is such a pair because 3=sigma(4)-4 and 4=sigma(3*3)-3*3.

Links

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

Crossrefs

Cf. A063990, A259180, A383239.

Programs

PARI

isok(k) = my(m = sigma(3*k) - 3*k); if (m>0, sigma(m) - m == k); \\ Michel Marcus, Apr 28 2025

Formula

We say that the numbers m and n form an (alpha, beta)-amicable pair if sigma(alpha*n)-alpha*n=m and sigma(beta*m)-beta*m=n, where alpha and beta are positive integers, and sigma(n) is the sum of the divisors of n.

Extensions

a(4)-a(6) from Michel Marcus, Apr 28 2025

A385586 Primes p such that there exists prime q < p such that sigma(p+1) = sigma(q+1) = p + q.

Original entry on oeis.org

37, 34687, 65587, 2089951, 8161477, 8340613, 18927067, 25855567, 64346413, 95150203, 238973101, 257658061, 277743397, 322210813, 349883707, 578403913, 704710543, 1121445337, 1654635937, 1741780693, 1804380007, 1963734061, 2346701941, 2360966173, 2720420707, 3232299517, 4343250181, 4925742973, 8085909913, 9044601133

Offset: 1

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Author

S. I. Dimitrov, Jul 03 2025

Keywords

nonn

Comments

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

Examples

(23, 37) is such a pair because sigma(23+1)=sigma(37+1) = 23 + 37.

Links

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

Crossrefs

Cf. A000203, A000040, A063990, A008333, A259180, A385718.

Subequence of A385705.

Programs

PARI

upto(n) = { res = List(); forprime(p = 2, n, s = sigma(p+1); q = s - p; if(q < p && isprime(q) && sigma(q+1) == s, print([p, q]); listput(res, p); ); ); res } \\ David A. Corneth, Jul 03 2025

Extensions

More terms from David A. Corneth, Jul 03 2025

A066873 Number of amicable pairs where smaller term of the pair is less than 10^n.

Original entry on oeis.org

0, 0, 1, 5, 13, 42, 108, 236, 586, 1427, 3340, 7642, 17519, 39374, 87102, 190775, 415523, 901312, 1947667, 4197267

Offset: 1

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Author

Shyam Sunder Gupta, Jan 21 2002

Keywords

nonn
more

Examples

There are 13 pairs with smaller term of the pair < 10^5, so a(5) = 13.

Links

Sergei Chernykh, Amicable pairs list.

Shyam Sunder Gupta, Amicable Numbers.

J. O. M. Pedersen, Known Amicable Pairs. [Broken link; WayBackMachine archive.]

J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link; WayBackMachine archive.]

J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]

Index entries for sequences related to amicable numbers

Crossrefs

Cf. A063990, A259180 (amicable pairs).

Extensions

Edited by Klaus Brockhaus, May 31 2003

a(15)-a(18) (from Sergei Chernykh's site) added by Amiram Eldar, Aug 19 2017

a(19) (from Sergei Chernykh's site) added by Amiram Eldar, Apr 05 2019

a(20) (from Sergei Chernykh's site) added by Amiram Eldar, Dec 13 2020

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A275316 Average of amicable pairs (x,y), ordered by the sum x+y given in A259953.

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A383239 Integers k such that there exists an integer 0

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A180202 The product of the two numbers in an amicable pair, A002025(n) * A002046(n).

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Mathematica

Formula

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

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Examples

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Formula

A262622 Amicable pairs of even numbers.

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PARI

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

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Examples

References

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Crossrefs

Programs

PARI

A307962 Lesser of coreful amicable numbers pair: numbers (m, n) such that csigma(m) = csigma(n) = m + n, where csigma(n) is the sum of the coreful divisors of n (A057723).

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Extensions

A383483 Numbers k such that k = sigma(m)-m where m = sigma(3*k)-3*k.

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A383483 Numbers k such that k = sigma(m)-m where m = sigma(3k)-3k.