A072890 -id:A072890 - cp's OEIS frontend

A003416 Sociable numbers: smallest member of each cycle (conjectured).

12496, 14316, 1264460, 2115324, 2784580, 4938136, 7169104, 18048976, 18656380, 28158165, 46722700, 81128632, 174277820, 209524210, 330003580, 498215416, 805984760, 1095447416, 1236402232, 1276254780, 1799281330

Offset: 1

N. J. A. Sloane

nonn

Numbers belonging to aliquot cycles of length greater than 2.
There is no proof that 564 (for example) is missing from this sequence (cf. A122726). - N. J. A. Sloane, Sep 17 2021
The lengths of the corresponding aliquot cycles are given by A052470. - Michel Marcus, Nov 15 2013

R. K. Guy, Unsolved Problems Number Theory, B7.
Paul Poulet, Parfaits, amiables et extensions, Editions Stevens, Bruxelles, 1918.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 91-92.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 174 Penguin Books 1987.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..53
Shyam Sunder Gupta, Perfect, Multiply Perfect, and Sociable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.
Verner E. Hoggart, Jr., Letters to N. J. A. Sloane, 1974-1975
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 112
David Moews, Perfect, amicable and sociable numbers
David Moews, A list of currently known aliquot cycles of length greater than 2 [This list is not known to be complete.]
Jan O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
Jan O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
Jan O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
Eric Weisstein's World of Mathematics, Sociable Numbers.

Cf. A052470, A072890, A072891, A122726.

Incorrect g.f. deleted by N. J. A. Sloane, Sep 20 2008

Added "conjectured" to definition. - N. J. A. Sloane, Sep 17 2021

A121507 Conjectured list of numbers whose aliquot sequence eventually reaches a cycle of length two or more.

Original entry on oeis.org

220, 284, 562, 1064, 1184, 1188, 1210, 1308, 1336, 1380, 1420, 1490, 1604, 1690, 1692, 1772, 1816, 1898, 2008, 2122, 2152, 2172, 2362, 2542, 2620, 2630, 2652, 2676, 2678, 2856, 2924, 2930, 2950, 2974, 3124, 3162, 3202, 3278, 3286, 3332, 3350, 3360

Offset: 1

Joshua Zucker, Aug 04 2006

nonn

For some numbers the outcome of the aliquot sequence is unknown. Currently, 276 is the least such.

Cf. A008892, A063769, A063990, A072890, A072891, A098007, A121508.

Edited by Don Reble, Aug 15 2006

A072891 The 5-cycle of the n => sigma(n)-n process, where sigma(n) is the sum of divisors of n (A000203).

Original entry on oeis.org

12496, 14288, 15472, 14536, 14264, 12496

Offset: 1

Miklos Kristof, Jul 29 2002

Called a "sociable" chain.

One of the two aliquot cycles of length greater than 2 that were discovered by Belgian mathematician Paul Poulet (1887-1946) in 1918 (the second is A072890). They were the only known such cycles until 1965 (see A072892). - Amiram Eldar, Mar 24 2024

Albert H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, New York: Dover Publications, 1964, Chapter IV, p. 28.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B7, p. 95.
Paul Poulet, La chasse aux nombres I: Parfaits, amiables et extensions, Bruxelles: Stevens, 1929.

Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.
Leonard Eugene Dickson, History of the Theory of Numbers, Vol. I: Divisibility and Primality, Washington, Carnegie Institution of Washington, 1919, p. 50.
Paul Poulet, Query 4865, L'Intermédiaire des Mathématiciens, Vol. 25 (1918), pp. 100-101.
Eric Weisstein's World of Mathematics, Sociable Numbers.
Wikipedia, Sociable number.

Cf. A000203, A001065, A003416, A072890, A072892.

NestWhileList[DivisorSigma[1, #] - # &, 12496, UnsameQ, All] (* Amiram Eldar, Mar 24 2024 *)

a(5+n) = a(n).

A072892 The 4-cycle of the n => sigma(n)-n process. sigma(n) is the sum of divisors of n. (A000203).

Original entry on oeis.org

1264460, 1547860, 1727636, 1305184, 1264460

Offset: 1

Miklos Kristof, Jul 29 2002

The two smallest members of sociable quadruples (1264460 and 2115324, see A090615) were found by the Canadian mathematician and educator Kenneth Dudley Fryer (1924-1984) in 1965 (Honsberger, 1970). These were the first aliquot cycles of length greater than 2 that were found since 1918 (see A072890 and A072891). They were rediscovered by Cohen (1970). - Amiram Eldar, Mar 24 2024

Ross Honsberger, Ingenuity in Mathematics, Mathematical Association of America, 1970.

Henri Cohen, On amicable and sociable numbers, Math. Comp., Vol. 24, No. 110 (1970), pp. 423-429.
Eric Weisstein's World of Mathematics, Sociable Numbers.
Wikipedia, Sociable number.

Cf. A000203, A001065, A072890, A072891, A090615.

NestWhileList[DivisorSigma[1, #] - # &, 1264460, UnsameQ, All] (* Amiram Eldar, Mar 24 2024 *)

a(4+n) = a(n).

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))

A347037 The length of the sequence before a repeated number appears, or -1 if a repeat never occurs, starting at k = n for the iterative cycle k -> sigma(k) - k if k is even, k -> sigma(k) if k is odd, where sigma(k) = sum of divisors of k.

Original entry on oeis.org

1, 2, 2, 2, 2, 1, 2, 2, 6, 3, 245, 244, 5, 4, 242, 243, 5, 4, 7, 6, 3, 5, 242, 241, 3, 244, 5, 1, 22, 21, 2, 8, 7, 8, 240, 7, 6, 6, 4, 21, 20, 4, 18, 245, 8, 7, 8, 4, 239, 246, 20, 19, 239, 5, 7, 3, 12, 11, 9, 8, 5, 5, 468, 18, 5, 4, 471, 6, 239, 238, 6, 5, 13, 6, 471, 17, 7, 6, 14, 5, 468

Offset: 1

Scott R. Shannon, Aug 12 2021

nonn

Various starting values, which are even or reach an even value, follow the associated trajectory of the standard aliquot sequence A098007 and thus grow to values whose ultimate fate is currently unknown. The first such term is 165 as its cycle begins 165 -> 288 -> 531 -> 780, from which point it follows the aliquot sequence starting at 780 that after 3485 further cycles reaches a term with 198 digits whose factorization is not known.
This sequence also has the same loops seen in the aliquot sequence. All even perfect numbers repeat after one value, and all even-numbered amicable pairs and even-numbered sociable number loops are also present. The amicable pair 2620-2934 is first reached from n = 477 after 329 cycles while the pair 1184-1210 is first reached from n = 847 after 4 cycles. The 28-member sociable number loop containing the number 376736, see A072890, is first reached from n = 267 after 52 cycles.
Loops present in this sequence that are not in the aliquot sequence are two-member loops formed by a Mersenne prime and the even number, a power of 2, one greater than it. For starting values n <= 1000 two other loops are present which end numerous cycles. One is a five-member loop comprising the numbers 56 -> 64 -> 63 -> 104 -> 106 (-> 56), first seen for starting value n = 11. The other is a four-member loop comprising the numbers 40 -> 50 -> 43 -> 44 (-> 40), first seen for starting value n = 27. It is likely other such loops exist for starting values >> 1000.
As sigma(prime) = prime + 1, such prime starting values will have the same cycle, with one extra term, as the later even term, assuming the prime is not a Mersenne prime.
For the first 1000 terms the longest currently known cycle if one of 1538 steps, starting at n = 693 and ending with 104. The largest value reached in this series is a 58-digit number 12316...65104. The majority of this series follows the 1074-step aliquot sequence starting at 1248.

			a(3) = 2 as 3 -> 4 -> 3. Similarly for all other Mersenne primes.
a(5) = 2 as 5 -> 6 -> 6. Similarly for all other primes one less than a perfect number.
a(6) = 1 as 6 -> 6. Similarly for all other even perfect numbers.
a(11) = 245 as 11 -> 12 -> 16 -> 15 -> 24 -> (233 more terms) -> 230 -> 202 -> 104 -> 106 -> 56 -> 64 -> 63 -> 104, ending with the five-member loop.
a(19) = 7 as 19 -> 20 -> 22 -> 14 -> 10 -> 8 -> 7 -> 8.
a(27) = 5 as 27 -> 40 -> 50 -> 43 -> 44 -> 40, ending with the four-member loop.
a(847) = 5 as 847 -> 1064 -> 1336 -> 1184 -> 1210 -> 1184, ending with the second smallest amicable pair.

Wikipedia, Aliquot sequence
P. Zimmermann, Aliquot Sequences

Cf. A098007, A001065, A259180, A003416, A072890, A000668.

Table[t=k=0;lst={n};k=If[OddQ@n,DivisorSigma[1,n],DivisorSigma[1,n]-n];While[FreeQ[lst,k],AppendTo[lst,k];n=k;t++;k=If[OddQ@n,DivisorSigma[1,n],DivisorSigma[1,n]-n]];t+1,{n,100}] (* Giorgos Kalogeropoulos, Aug 14 2021 *)

cp's OEIS Frontend

A003416 Sociable numbers: smallest member of each cycle (conjectured).

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A121507 Conjectured list of numbers whose aliquot sequence eventually reaches a cycle of length two or more.

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A072891 The 5-cycle of the n => sigma(n)-n process, where sigma(n) is the sum of divisors of n (A000203).

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A072892 The 4-cycle of the n => sigma(n)-n process. sigma(n) is the sum of divisors of n. (A000203).

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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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A347037 The length of the sequence before a repeated number appears, or -1 if a repeat never occurs, starting at k = n for the iterative cycle k -> sigma(k) - k if k is even, k -> sigma(k) if k is odd, where sigma(k) = sum of divisors of k.

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