A063990 -id:A063990 - cp's OEIS frontend

A260087 Larger of amicable pair (x, y) as they are listed in A259933.

284, 1210, 2924, 5564, 6368, 10856, 14595, 18416, 66992, 71145, 76084, 87633, 88730, 124155, 123152, 139815, 153176, 168730, 176336, 180848, 203432, 202444, 365084, 389924, 430402, 399592, 455344, 486178, 514736, 525915, 669688, 686072, 652664, 691256, 712216, 783556, 796696, 863835, 901424, 980984, 1043096, 1125765

Offset: 1

Omar E. Pol, Jul 15 2015

nonn

Another version of A002046.

First differs from A002046 at a(9).

Laszlo Hars, Performance compared to mathematica Julia-users (2014)
Khelleos, Amicable numbers, CyberForum.ru (2011)
OEIS Wiki, Amicable numbers (This page needs work)
Wikipédia, Barátságos számok (contains a mistake: A063990 should be replaced with A259933)

Cf. A002046, A063990, A259180, A259933, A259953, A260086.

a(n) = A259933(2n) = A259953(n) - A259933(2n-1) = A259953(n) - A260086(n).

A262623 Amicable pairs of odd numbers.

Original entry on oeis.org

12285, 14595, 67095, 71145, 69615, 87633, 100485, 124155, 122265, 139815, 522405, 525915, 802725, 863835, 947835, 1125765, 1175265, 1438983, 1280565, 1340235, 1358595, 1486845, 1798875, 1870245, 4482765, 5120595, 5357625, 5684679, 5730615, 6088905, 6377175, 6680025, 8619765, 9627915, 9071685, 9498555, 9206925, 10791795

Offset: 1

Omar E. Pol, Nov 09 2015

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

First differs from A262625 at a(4).

Song Y. Yan, Perfect, Amicable and Sociable Numbers. A Computational Approach, World Scientific, 1996, pages 151 - 153.

Amiram Eldar, Table of n, a(n) for n = 1..20000
Sergei Chernykh, Amicable pairs list

Cf. A002025, A002046, A005408, A063990, A259180, A262622, A262625.

lista(nn) = {forstep(n=1, nn, 2, m = sigma(n)-n; if ((m > n) && (n==sigma(m)-m), print1(n, ", ", m, ", ")););} \\ Michel Marcus, Nov 14 2015

A038182 3-infinitary perfect numbers k: 3-i-sigma(k) = 2*k, where 3-i-sigma = A049418.

Original entry on oeis.org

6, 28, 3024, 6552, 27578880, 49266240, 49095705098695680

Offset: 1

Yasutoshi Kohmoto

Similarly, we have 3-i-sigma(x)/x = r for the following numbers: r = 3 for x = 672, 13104, 4021920, 55157760, 98532480, 459818240, 372667889664, 7267023848448, 1178296922368696320, 5498718971053916160, ...; r = 4 for x = 2178540; r = 3/2 for x = 2, 24, 9192960, 196382820394782720. (Values above 10^7 from Yasutoshi Kohmoto, some terms may be missing.) - M. F. Hasler, Sep 21 2022

			Factorizations: 2*3, 2^2*7, 2^4*3^3*7, 2^3*3^2*7*13, 2^9*3^4*5*7*19, 2^6*3*5*19*37*73, 2^10*3^6*5*19^2*127*379*757.

J. O. M. Pedersen, Tables of Aliquot Cycles: backup on web.archive.org of no more existing web page, as of May 2014.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]

Cf. A007357, A037445, A038148, A049418, A074849, A097464.

f[p_, e_] := Module[{d = IntegerDigits[e, 3]}, m = Length[d]; Product[(p^((d[[j]] + 1)*3^(m - j)) - 1)/(p^(3^(m - j)) - 1), {j, 1, m}]]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[7000], s[#] == 2*# &] (* Amiram Eldar, Oct 24 2024 *)

is_A038182(n)=A049418(n)==2*n \\ M. F. Hasler, Sep 21 2022

Definition shortened by R. J. Mathar, Oct 06 2010

A126016 Numbers whose aliquot sequence does not terminate in 1.

Original entry on oeis.org

6, 25, 28, 95, 119, 143, 220

Offset: 1

Franklin T. Adams-Watters, Dec 14 2006

Sequence continues 276?, 284, 306?, 396?, 417, 445, 496, .... Because 276, 306 and 396 are all in the same family, either all 3 are present or none are. It is not known whether any aliquot sequence grows without bound; 276 is the smallest number for which this is unknown.
Additional tentative terms: 552, 562, 564, 565, 608, 650, 652, 660, 675, 685, 696, 780, 783, 790, 828, 840, 888, 909, 913, 966, 996, 1064, 1074, 1086, 1098, ... - Jean-François Alcover, Nov 14 2013
For additional terms, if the Goldbach Conjecture is assumed, take any odd term, subtract 1, and find two distinct primes that sum to it. For some numbers there will not be any pair of distinct primes. Multiply the two primes and the product is an element of the sequence. Note that this process does not work if the term - 1 is power of a prime. - Nathaniel J. Strout, Nov 25 2018

Eric Weisstein's World of Mathematics, Aliquot Sequence
P. Zimmermann, Latest information

Complement of A080907. Includes A000396, A063990 and other sociable numbers, A063769, numbers whose aliquot sequence reaches a sociable number and numbers whose aliquot sequence grows without bound.

maxAliquot = 10^45; A131884 = {}; s[1] = 1; s[n_] := DivisorSigma[1, n] - n; selQ[n_ /; n <= 5] = True; selQ[n_] := NestWhile[s, n, If[{##}[[-1]] > maxAliquot, Print["A131884: ", n]; AppendTo[A131884, n]; False, Length[{##}] < 4 || {##}[[-4 ;; -3]] != {##}[[-2 ;; -1]]] & , All] == 1; Reap[For[k = 1, k < 1100, k++, If[!selQ[k], Print[k]; Sow[k]]]][[2, 1]]

A127660 Integers whose exponential aliquot sequences end in an exponential amicable pair.

Original entry on oeis.org

90972, 100548, 454860, 502740, 937692, 968436, 1000692, 1106028, 1182636, 1307124, 1383732, 1536416, 1546524, 1709316, 2092356, 2312604, 2502528, 2638188, 2690100, 2820132, 2915892, 3116988, 3365964, 3720276, 3729852, 3907008, 3911796, 4122468, 4248552, 4275684

Offset: 1

Ant King, Jan 25 2007

nonn

Sometimes called the exponential 2-cycle attractor set. The first 10 terms of this sequence are the same as the first 10 terms of A127659.

			a(11) = 1383732 because the eleventh integer whose exponential aliquot sequence ends in an exponential amicable pair is 1383732.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, Jr., Some results concerning exponential divisors, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.
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]

Cf. A127656, A127657, A127658.

Subsequences: A127659, A126165, A126166.

ExponentialDivisors[1]={1};ExponentialDivisors[n_]:=Module[{}, {pr,pows}=Transpose@FactorInteger[n];divpowers=Distribute[Divisors[pows],List];Sort[Times@@(pr^Transpose[divpowers])]];se[n_]:=Plus@@ExponentialDivisors[n]-n;g[n_] := If[n > 0, se[n], 0];eTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];ExponentialAmicableNumberQ[k_]:=If[Nest[se,k,2]==k && !se[k]==k,True,False];Select[Range[5 10^6],ExponentialAmicableNumberQ[Last[eTrajectory[ # ]]] &]
f[p_, e_] := DivisorSum[e, p^# &]; s[0] = s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; q[n_] := Module[{v = NestWhileList[s, n, UnsameQ, All]}, v[[-2]] != v[[-1]] > 0 && v[[-3]] == v[[-1]]]; Select[Range[10^6], q] (* Amiram Eldar, Mar 11 2023 *)

A127667 Odd integers that do not generate monotonically decreasing infinitary aliquot sequences.

Original entry on oeis.org

945, 1743, 2175, 2655, 2823, 2865, 3105, 3375, 3537, 3585, 3729, 4209, 4665, 5775, 6559, 6681, 6969, 7257, 7263, 7785, 8457, 8583, 9657, 10017, 10047, 10113, 10395, 10599, 10743, 12285, 13815, 14055, 14145, 15015, 15597, 16065, 17955, 18529, 18777, 19305, 19635

Offset: 1

Ant King, Jan 26 2007

nonn

Based on empirical evidence, approximately 98.9 % of the infinitary aliquot sequences generated by the odd integers are monotonically decreasing. This sequence represents the 1.1 % of odd integers that are the exceptions to this.

			a(5)=2823 because 2823 is the fifth odd integer whose infinitary aliquot sequence is not monotonically decreasing.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
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]

Cf. A126168, A127661, A127666.

ExponentList[n_Integer,factors_List]:={#,IntegerExponent[n,# ]}&/@factors;InfinitaryDivisors[1]:={1}; InfinitaryDivisors[n_Integer?Positive]:=Module[ { factors=First/@FactorInteger[n], d=Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f,g}, BitOr[f,g]==g][ #,Last[ # ]]]&/@ Transpose[Last/@ExponentList[ #,factors]&/@d]],?(And@@#&),{1}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;g[n_] := If[n > 0,properinfinitarydivisorsum[n], 0];iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];u[n_]:=Table[n[[k+1]]

More terms from Amiram Eldar, Sep 16 2019

A206708 Numbers k such that sigma(k) = sigma(sigma(k)-k).

Original entry on oeis.org

6, 28, 220, 284, 496, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 8128, 10744, 10856, 12285, 14595, 17296, 18416, 63020, 66928, 66992, 67095, 69615, 71145, 76084, 79750, 87633, 88730, 100485, 122265, 122368, 123152, 124155, 139815, 141664, 142310, 153176

Offset: 1

Michel Lagneau, Feb 11 2012

nonn

For all k, let s(k) = sigma(k) - k, the aliquot sum function A001065; then this sequence is the set of k such that s(s(k)) = k. - Jeppe Stig Nielsen, Jan 12 2020

			220 is in the sequence because sigma(220) = 504, sigma(504 - 220) = sigma(284) = 504.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Amiram Eldar)
Eric Weisstein's World of Mathematics, Perfect Number
Eric Weisstein's World of Mathematics, Amicable Pair
Wikipedia, Perfect number
Wikipedia, Amicable number

Cf. A000396 (perfect numbers), A063990 (amicable numbers).

Cf. A000203 (sum of divisors), A001065 (sum of proper divisors).

[k:k in [2..154000]|s eq DivisorSigma(1,s-k) where s is DivisorSigma(1,k)]; // Marius A. Burtea, Jan 13 2020

q:= n-> (s-> s(n)=s(s(n)-n))(numtheory[sigma]):
select(q, [$1..100000])[];  # Alois P. Heinz, Jan 31 2023

Select[Range[10^6],DivisorSigma[1,#]==DivisorSigma[1, DivisorSigma[1,#]-#]&]

isok(k) = if (k != 1, my(sk=sigma(k)); sk == sigma(sk-k)); \\ Michel Marcus, Jun 24 2019

Equals {A063990} union {A000396} = (amicable numbers) union (perfect numbers).

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

A383484 Integers k such that there exists an integer 0

Original entry on oeis.org

3, 33, 69, 77, 133, 161, 235, 267, 287, 321, 385, 699, 715, 1235, 1379, 1437, 1529, 1595, 1653, 1719, 2047, 2233, 2241, 2569, 2727, 2829, 3237, 3269, 3999, 4585, 4683, 4911, 5075, 5163, 5215, 5497, 5667, 5691, 7085, 7089, 7587, 7761, 7797, 7945, 8259, 9159, 9659, 10653

Offset: 1

S. I. Dimitrov, Apr 28 2025

nonn

From David A. Corneth, May 04 2025: (Start)
If (t, u) is a divisor pair of sigma(k)^2 then m = (t + u - 2*k)/2, sigma(m) = m + k - t.
Proof:
Since sigma(m)^2 + sigma(k)^2 = (m+k)^2 we have sigma(k)^2 = (m+k)^2 - sigma(m)^2 = (m + k - sigma(m)) * (m + k + sigma(m)) = t * u where t, u | sigma(k)^2.
This gives the system (m + k - sigma(m)) = t and (m + k + sigma(m)) = u. Solving gives
m = (t + u - 2*k)/2, sigma(m) = m + k - t. For every pair (t, u) of divisors of sigma(k)^2 we can test if the given values of m and sigma(m) hold. If at least one of them holds then k is in the sequence. Q. E. D.
Are there any even terms? There are none in the first 1006 terms. (End)

			(2, 3) is such a pair because sigma^2(2)+sigma^2(3) = 3^2+4^2 = (2+3)^2.
33 is in the sequence. As sigma(33)^2 = 2304 and for the divisor pair (32, 72) we have m = (32 + 72 - 2*33)/2 = 19 and sigma(m) = m + k - 32 = 19 + 33 - 32 = 20 and indeed sigma(19) = 20. - _David A. Corneth_, May 04 2025

David A. Corneth, Table of n, a(n) for n = 1..2003
S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Cf. A063990, A259180, A383239.

isok(k) = for (m=1, k, if (sigma(m)^2+sigma(k)^2==(m+k)^2, return(1))); \\ Michel Marcus, Apr 28 2025

is(n) = {my(sn = sigma(n)^2, d = divisors(sn)); for(i = 1, #d / 2, k = (d[i] + d[#d + 1 - i] - 2*n) / 2; if(denominator(k) == 1, sk = n + k - d[i]; if(k < n && sigma(k) == sk, return(1)))); 0} \\ David A. Corneth, May 04 2025

More terms from Michel Marcus, Apr 28 2025

A125601 a(n) is the smallest k > 0 such that there are exactly n numbers whose sum of proper divisors is k.

Original entry on oeis.org

2, 3, 6, 21, 37, 31, 49, 79, 73, 91, 115, 127, 151, 121, 181, 169, 217, 265, 253, 271, 211, 301, 433, 379, 331, 361, 457, 391, 451, 655, 463, 541, 421, 775, 511, 769, 673, 715, 865, 691, 1015, 631, 1069, 1075, 721, 931, 781, 1123, 871, 925, 901, 1177, 991, 1297

Offset: 0

Klaus Brockhaus, Nov 27 2006

nonn

Minimal values for nodes of exact degree in aliquot sequences. Find each node's degree (number of predecessors) in aliquot sequences and choose the smallest value as the sequence member. - Ophir Spector, ospectoro (AT) yahoo.com Nov 25 2007

			a(4) = 37 since there are exactly four numbers (155, 203, 299, 323) whose sum of proper divisors is 37. For k < 37 there are either fewer or more numbers (32, 125, 161, 209, 221 for k = 31) whose sum of proper divisors is k.

Daniel Mondot, Table of n, a(n) for n = 0..5646 (First 157 terms from Ophir Spector. First 1000 terms from Ophir Spector and Donovan Johnson.)
W. Creyaufmueller, Aliquot sequences
Daniel Mondot, A-file, extended version of b-file but with some gaps towards the end.
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]
Eric Weisstein's World of Mathematics, Aliquot sequence

Cf. A001065, A048138, A070015, A123930, A080907, A115350, A121507, A037020, A126016, A057709, A057710, A063990.

{m=54;z=1500;y=600000;v=vector(z);for(n=2,y,s=sigma(n)-n; if(s

cp's OEIS Frontend

A260087 Larger of amicable pair (x, y) as they are listed in A259933.

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A262623 Amicable pairs of odd numbers.

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A038182 3-infinitary perfect numbers k: 3-i-sigma(k) = 2*k, where 3-i-sigma = A049418.

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Mathematica

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A126016 Numbers whose aliquot sequence does not terminate in 1.

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Mathematica

A127660 Integers whose exponential aliquot sequences end in an exponential amicable pair.

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Mathematica

A127667 Odd integers that do not generate monotonically decreasing infinitary aliquot sequences.

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Mathematica

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A206708 Numbers k such that sigma(k) = sigma(sigma(k)-k).

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Magma

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

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