A072891 -id:A072891 - cp's OEIS frontend

A098008 Length of transient part of aliquot sequence for n, or -1 if transient part is infinite.

1, 2, 2, 3, 2, 0, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, 4, 2, 7, 3, 6, 2, 5, 1, 7, 3, 0, 2, 15, 2, 3, 6, 8, 3, 4, 2, 7, 3, 4, 2, 14, 2, 5, 7, 8, 2, 6, 4, 3, 4, 9, 2, 13, 3, 5, 3, 4, 2, 11, 2, 9, 3, 4, 3, 12, 2, 5, 4, 6, 2, 9, 2, 5, 5, 5, 3, 11, 2, 7, 5, 6, 2, 6, 3, 9, 7, 7, 2, 10, 4, 6, 4, 4, 2, 9, 2, 3, 4, 5, 2, 18

Offset: 1

N. J. A. Sloane, Sep 09 2004

See A098007 for further information.
a(n) = 0 if and only if n is perfect (A000396) or part of a cycle of length greater than 1. - Comment corrected by Antti Karttunen, Nov 02 2017.
It is believed that the first time a(n) = -1 is at n = 276 (see A008892). - N. J. A. Sloane, Nov 02 2017

			From _Antti Karttunen_, Nov 02 2017: (Start)
For n = 3, a(n) = 2, because A001065(3) = 1 and A001065(1) = 0, so it took two steps to end in zero.
For n = 25, a(n) = 1, because A001065(25) = 6, and A001065(6) = 6, so it took one step to enter into a cycle.
For n = 12496, a(n) = 0, because 12496 is a member of 5-cycle of map n -> A001065(n) (see A072891).
(End)

R. K. Guy, Unsolved Problems in Number Theory, B6.
R. K. Guy and J. L. Selfridge, Interim report on aliquot series, pp. 557-580 of Proceedings Manitoba Conference on Numerical Mathematics. University of Manitoba, Winnipeg, Oct 1971.

Antti Karttunen, Table of n, a(n) for n = 1..275

Cf. A001065, A098007, A044050, A003023, A008892. See A007906 for another version.

Cf. A206708 (gives a proper subset of zeros).

g[n_] := If[n > 0, DivisorSigma[1, n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; Table[ Length[ f[n]] - 2, {n, 102}] (* good only for n<220 *) (* Robert G. Wilson v, Sep 10 2004 *)

(define (A098008 n) (let loop ((visited (list n)) (i 1)) (let ((next (A001065 (car visited)))) (cond ((zero? next) i) ((member next visited) => (lambda (transientplus1) (- (length transientplus1) 1))) (else (loop (cons next visited) (+ 1 i))))))) ;; Good for at least n = 1..275.
(define (A001065 n) (- (A000203 n) n)) ;; For an implementation of A000203, see under that entry.
;; Antti Karttunen, Nov 02 2017

More terms from Robert G. Wilson v, Sep 10 2004

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

Original entry on oeis.org

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

A072890 The 28-cycle of the n => sigma(n)-n process, where sigma(n) is the sum of divisors of n (A000203).

Original entry on oeis.org

14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716, 14316

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 A072891). 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, pp. 28-29.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B7, p. 95.
C. Stanley Ogilvy, Tomorrow's math, unsolved problems for the amateur,Oxford University Press, 2nd ed., 1972, p. 113.
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, A072891, A072892.

NestList[DivisorSigma[1,#]-#&,14316,28] (* Harvey P. Dale, Oct 27 2013 *)

a(28+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))

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A098008 Length of transient part of aliquot sequence for n, or -1 if transient part is infinite.

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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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A072890 The 28-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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