A126016 -id:A126016 - cp's OEIS frontend

A080907 Numbers whose aliquot sequence terminates in a 1.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Gabriel Cunningham (gcasey(AT)mit.edu), Mar 31 2003

All primes are in this set because s(p) = 1 for p prime. Perfect numbers are clearly not in this set. Neither are aspiring numbers (A063769), or numbers whose aliquot sequence is a cycle (such as 220 and 284).

There are some numbers whose aliquot sequences haven't been fully determined (such as 276).

			4 is in this set because its aliquot chain is 4->3->1. 6 is not in this set because it is perfect. 25 is not in this set because its aliquot chain is 25->6.

Jean-François Alcover, Table of n, a(n) for n = 1..1000
M. Benito, W. Creyaufmüller, J. L. Varona, and P. Zimmermann, Aliquot Sequence 3630 Ends After Reaching 100 Digits, Experimental Mathematics (2002), Volume 11, issue 2.
Markus Tervooren, Aliquot sequence of 840
Eric Weisstein's World of Mathematics, Aliquot Sequence

Cf. A063769, A115350, A131884.

Complement of A126016.

maxAliquot = 10^50; 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; Select[Range[1, 1100], selQ] (* Jean-François Alcover, Nov 14 2013, updated Sep 10 2015 *)

n is a member if n = 1 or s(n) is a member, where s(n) is the sum of the proper factors of n.

Edited by N. J. A. Sloane, Aug 14 2006
More terms from Franklin T. Adams-Watters, Dec 14 2006
The fact that 840 was missing from the sequence b-file was pointed out by Philip Turecek, Sep 10 2015

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

A135244 Largest m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

Original entry on oeis.org

0, 4, 9, 0, 25, 8, 49, 15, 14, 21, 121, 35, 169, 33, 26, 55, 289, 77, 361, 91, 38, 85, 529, 143, 46, 133, 28, 187, 841, 221, 961, 247, 62, 253, 24, 323, 1369, 217, 81, 391, 1681, 437, 1849, 403, 86, 493, 2209, 551, 94, 589, 0, 667, 2809, 713, 106, 703, 68, 697, 3481

Offset: 2

Ophir Spector (ospectoro(AT)yahoo.com), Nov 25 2007

nonn

Previous name: Aliquot predecessors with the largest values.
Find each node's predecessors in aliquot sequences and choose the largest predecessor.
Climb the aliquot trees on shortest paths (see A135245 = Climb the aliquot trees on thickest branches).
The sequence starts at offset 2, since all primes satisfy sigma(n)-n = 1. - Michel Marcus, Nov 11 2014

			a(25) = 143 since 25 has 3 predecessors (95,119,143), 143 being the largest.
a(5) = 0 since it has no predecessors (see Untouchables - A005114).

Amiram Eldar, Table of n, a(n) for n = 2..10000 (terms 2..150 from Ophir Spector)
Wolfgang Creyaufmueller, Aliquot sequences.
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, A005114, A125601, A135245, A057709, A057710, A063769, A080907, A121507, A037020, A126016.

seq[max_] := Module[{s = Table[0, {n, 1, max}], i}, Do[If[(i = DivisorSigma[1, n] - n) <= max, s[[i]] = Max[s[[i]], n]], {n, 2, (max - 1)^2}]; Rest @ s]; seq[50]

lista(nn) = {for (n=2, nn, k = (n-1)^2; while(k && (sigma(k)-k != n), k--); print1(k, ", "););} \\ Michel Marcus, Nov 11 2014

a(1)=0 removed and offset set to 2 by Michel Marcus, Nov 11 2014

New name from Michel Marcus, Oct 31 2023

A135245 Aliquot predecessors with the largest degrees.

Original entry on oeis.org

0, 0, 4, 9, 0, 25, 8, 49, 15, 14, 21, 121, 35, 169, 33, 12, 55, 289, 65, 361, 91, 20, 85, 529, 143, 46, 133, 28, 187, 841, 161, 961, 247, 62, 253, 24, 323, 1369, 217, 81, 391, 1681, 341, 1849, 403, 86, 493, 2209, 551, 40, 481, 0, 667, 2809, 533, 106, 703, 68, 697, 3481

Offset: 1

Ophir Spector, ospectoro (AT) yahoo.com, Nov 25 2007

nonn

Find each node's predecessors in aliquot sequences and choose the node with largest number of predecessors.

Climb the aliquot trees on thickest branches (see A135244 = Climb the aliquot trees on shortest paths).

			a(25) = 143 since 25 has 3 predecessors (95,119,143) with degrees (4,5,7), 143 having the largest degree. a(5) = 0 since it has no predecessors (see Untouchables - A005114).

Ophir Spector, Table of n, a(n) for n = 1..150
W. Creyaufmueller, Aliquot sequences
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 A005114 A125601 A135244 A057709 A057710 A063769 A080907 A121507 A037020 A126016.

A292217 Conjectured list of numbers in increasing order that belong to sociable cycles of length greater than 2 in which the sum of the cycle is divisible by 10.

Original entry on oeis.org

1264460, 1305184, 1547860, 1727636, 4938136, 5423384, 5504056, 5753864, 18656380, 20522060, 24289964, 28158165, 28630036, 29902635, 29971755, 30853845, 81128632, 91314968, 91401368, 96389032, 209524210, 230143790, 231439570, 246667790, 498215416, 506040584, 510137384, 583014136

Offset: 1

Zoltan Galantai, Sep 11 2017

This list is not known to be complete (564 might be a member). See A122726. - N. J. A. Sloane, Sep 17 2021

Up to the known 1593 sociable number cycles, 96.1% of the sociable number cycles satisfy this condition (up to the first 10 sociable number cycles: 40%; up to the first 100 sociable number cycles: 77%; up to the first 500 sociable number cycles: 92%, and up to the first 1000 sociable number cycles: 94.9%). So the conjecture here is that as the number of sociable number cycles increases, the percentage of the sums of the sociable number cycles divisible by 10 approaches 100%. Notice that the sums of amicable pairs are similarly often divisible by 10, but are not included here (see A291422).

			The sum of 1264460, 1547860, 1727636 and 1305184 is divisible by ten, thus this sociable number cycle belongs to the sequence. On the other hand, the 12496, 14288, 15472, 14536, 14264 sociable number cycle does not qualify since its sum is 71506.

R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1994, pp. 62 - 63.
Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chappman and HALL/CRC, 2003, pp. 2747 - 2748.
Song Y. Yan, Perfect, Amicable and Sociable Numbers. A Computation Approach, World Scientific 1996, pp. 34 - 38.

Zoltan Galantai, Table of n, a(n) for n = 1..36
Zoltan Galantai, List of known sociable number cycles where the sums of the cycles is divisible by 10.
Shyam Sunder Gupta, Perfect, Multiply Perfect, and Sociable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.
David Moews, A list of currently known aliquot cycles of length greater than 2 [This list is not known to be complete.]
Eric Weisstein's World of Mathematics, Sociable Numbers

Cf. A122726, A003416, A090615, A119478, A098007, A126016, A063990, A291422.

Changed definition and added comment to point out that this sequence is only conjectural. - N. J. A. Sloane, Sep 17 2021

A347769 a(0) = 0; a(1) = 1; for n > 1, a(n) = A001065(a(n-1)) = sigma(a(n-1)) - a(n-1) (the sum of aliquot parts of a(n-1)) if this is not yet in the sequence; otherwise a(n) is the smallest number missing from the sequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 15, 13, 14, 17, 18, 21, 19, 20, 22, 23, 24, 36, 55, 25, 26, 27, 28, 29, 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 31, 32, 34, 35, 37, 38, 39, 40, 50, 43, 41, 44, 46, 47, 48, 76, 64, 63, 49, 51, 52, 53, 56, 57, 58, 59, 60, 108, 172

Offset: 0

Eric Chen, Sep 13 2021

nonn

This sequence is a permutation of the nonnegative integers iff Catalan's aliquot sequence conjecture (also called Catalan-Dickson conjecture) is true.
a(563) = 276 is the smallest number whose aliquot sequence has not yet been fully determined.
As long as the aliquot sequence of 276 is not known to be finite or eventually periodic, a(563+k) = A008892(k).

			a(0) = 0, a(1) = 1;
since A001065(a(1)) = 0 has already appeared in this sequence, a(2) = 2;
since A001065(a(2)) = 1 has already appeared in this sequence, a(3) = 3;
...
a(11) = 11;
since A001065(a(11)) = 1 has already appeared in this sequence, a(12) = 12;
since A001065(a(12)) = 16 has not yet appeared in this sequence, a(13) = A001065(a(12)) = 16;
since A001065(a(13)) = 15 has not yet appeared in this sequence, a(14) = A001065(a(13)) = 15;
since A001065(a(14)) = 9 has already appeared in this sequence, a(15) = 13;
...

Eric Chen, Table of n, a(n) for n = 0..2703 (all currently known terms, after the b-file of A008892)
Eric Weisstein's World of Mathematics, Aliquot sequence
Eric Weisstein's World of Mathematics, Catalan's Aliquot Sequence Conjecture
Wikipedia, Aliquot sequence

Cf. A032451.
Cf. A001065 (sum of aliquot parts).
Cf. A003023, A044050, A098007, A098008: ("length" of aliquot sequences, four versions).
Cf. A007906.
Cf. A115060 (maximum term of aliquot sequences).
Cf. A115350 (termination of the aliquot sequences).
Cf. A098009, A098010 (records of "length" of aliquot sequences).
Cf. A290141, A290142 (records of maximum term of aliquot sequences).
Cf. A000396, A063990, A122726, A206708, A347770.
Cf. A080907, A063769, A121507, A121508, A131884, A126016.
Aliquot sequences starting at various numbers: A000004 (0), A000007 (1), A033322 (2), A010722 (6), A143090 (12), A143645 (24), A010867 (28), A008885 (30), A143721 (38), A008886 (42), A143722 (48), A143723 (52), A008887 (60), A143733 (62), A143737 (68), A143741 (72), A143754 (75), A143755 (80), A143756 (81), A143757 (82), A143758 (84), A143759 (86), A143767 (87), A143846 (88), A143847 (96), A143919 (100), A008888 (138), A008889 (150), A008890 (168), A008891 (180), A203777 (220), A008892 (276), A014360 (552), A014361 (564), A074907 (570), A014362 (660), A269542 (702), A045477 (840), A014363 (966), A014364 (1074), A014365 (1134), A074906 (1521), A143930 (3630), A072891 (12496), A072890 (14316), A171103 (46758), A072892 (1264460).

A347769_list(N)=print1(0, ", "); if(N>0, print1(1, ", ")); v=[0, 1]; b=1; for(n=2, N, if(setsearch(Set(v), sigma(b)-b), k=1; while(k<=n, if(!setsearch(Set(v), k), b=k; k=n+1, k++)), b=sigma(b)-b); print1(b, ", "); v=concat(v, b))

cp's OEIS Frontend

A080907 Numbers whose aliquot sequence terminates in a 1.

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

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A135244 Largest m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

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A135245 Aliquot predecessors with the largest degrees.

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A292217 Conjectured list of numbers in increasing order that belong to sociable cycles of length greater than 2 in which the sum of the cycle is divisible by 10.

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A347769 a(0) = 0; a(1) = 1; for n > 1, a(n) = A001065(a(n-1)) = sigma(a(n-1)) - a(n-1) (the sum of aliquot parts of a(n-1)) if this is not yet in the sequence; otherwise a(n) is the smallest number missing from the sequence.

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