A052470 -id:A052470 - cp's OEIS frontend

A090615 Smallest member of sociable quadruples.

1264460, 2115324, 2784580, 4938136, 7169104, 18048976, 18656380, 28158165, 46722700, 81128632, 174277820, 209524210, 330003580, 498215416, 1236402232, 1799281330, 2387776550, 2717495235, 2879697304, 3705771825, 4424606020, 4823923384, 5373457070, 8653956136

Offset: 1

Eric W. Weisstein, Dec 06 2003

nonn

120 sociable numbers of order 4 are known as of Feb. 2003.
142 were known in 2007 (http://amicable.homepage.dk/knwnc4.htm).
201 are known in 2012.
210 were known in April 2013. - Michel Marcus, Nov 10 2013
From Amiram Eldar, Mar 24 2024: (Start)
The terms were found by:
  a(1)-a(2) - Kenneth Dudley Fryer in 1965 (Honsberger, 1970; see also A072892)
  a(3)-a(7), a(9) - Cohen (1970)
  a(8) - Borho (1969)
  a(10)-a(13) - independently by Richard David (1972; Devitt et al., 1976, Guy 1977) and Steve C. Root (Beeler et al. 1972)
  a(14) - Steve C. Root in 1972
  a(15)-a(22) - Flammenkamp (1991)
  a(23)-a(24) - Moews and Moews (1991)
  a(25)-a(27) - Moews and Moews (1993)
(End)

Walter Borho, Über die Fixpunkte der k-fach iterierten Teilersummenfunktion, Mitt. Math. Gesellsch. Hamburg, Vol. 9, No. 5 (1969), pp. 34-48.
Richard David, Letter to D. H. Lehmer, February 25 , 1972.
John Stanley Devitt, Richard K. Guy, and John L. Selfridge, Third report on aliquot sequences, Proceedings of the Sixth Manitoba Conference on Numerical Mathematics, September 29 - October 2, 1976, Congressus Numerantium XVIII, University of Manitoba, Winnipeg, Manitoba, Utilitas Mathematics Publications, 1976, pp. 177-204.
Richard K. Guy, "Aliquot Sequences", in: Hans Zassenhaus (ed.), Number Theory and Algebra: Collected Papers Dedicated to Henry B. Mann, Arnold E. Ross, and Olga Taussky-Toddm, Academic Press Inc., 1977.
Ross Honsberger, Ingenuity in Mathematics, Mathematical Association of America, 1970.

Amiram Eldar, Table of n, a(n) for n = 1..48
Michael Beeler, R. William Gosper and Richard C. Schroeppel, HAKMEM, MIT Artificial Intelligence Laboratory report AIM-239, Feb. 29, 1972, Item 61; HTML transcription.
Karsten Blankenagel, Walter Borho, and Axel vom Stein, New amicable four-cycles, Math. Comp., Vol. 72, No. 244 (2003), pp. 2071-2076.
Karsten Blankenagel and Walter Borho, New amicable four-cycles II, International Journal of Mathematics and Scientific Computing, Vol. 5, No. 1 (2015), pp. 49-51.
Henri Cohen, On amicable and sociable numbers, Math. Comp., Vol. 24, No. 110 (1970), pp. 423-429.
Achim Flammenkamp, New sociable numbers, Mathematics of Computation, Vol. 56, No. 194 (1991), pp. 871-873.
Richard K. Guy and John L. Selfridge, Interim report on aliquot series, pp. 557-580 of Proceedings Manitoba Conference on Numerical Mathematics. University of Manitoba, Winnipeg, Oct 1971.
David Moews, A list of currently known aliquot cycles of length greater than 2.
David Moews and Paul C. Moews, Aliquot cycles below 10^10, Mathematics of Computation, Volume 57, No. 196 (1991), pp. 849-855.
David Moews and Paul C. Moews, A search for aliquot cycles and amicable pairs, Mathematics of computation, Vol. 61, No. 204 (1993), pp. 935-938.
Jan Munch Pedersen, Known Sociable Numbers of order four, Tables of Aliquot Cycles.
Eric Weisstein's World of Mathematics, Sociable Numbers.
Wikipedia, Sociable number.

Cf. A003023, A003416, A052470, A072892.

a(22)-a(24) from Flammenkamp (1991) and Moews and Moews (1991) added by Amiram Eldar, Mar 24 2024

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

A122726 Conjectured list of sociable numbers.

Original entry on oeis.org

12496, 14264, 14288, 14316, 14536, 15472, 17716, 19116, 19916, 22744, 22976, 31704, 45946, 47616, 48976, 83328, 97946, 122410, 152990, 177792, 243760, 274924, 275444, 285778, 294896, 295488, 358336, 366556, 376736, 381028, 418904, 589786

Offset: 1

Tanya Khovanova, Sep 23 2006

nonn

Comments from David Moews, Sep 17 2021: (Start)
It is possible that there are quite small numbers missing from this sequence. There is no proof that 564 (for example) is missing.
Let s(n) = sigma(n)-n denote the sum of the divisors of n, excluding n itself. The aliquot sequence starting at n is the sequence n, s(n), s(s(n)), s(s(s(n))), ...
Starting at 564, the aliquot sequence continues for at least 3486 steps, reaching a 198-digit number after 3486 iterations of s.  In the reverse direction, 563 = 564 - 1 is prime so s(563^2) = 564 and also s(7*316961) = 563^2, s(17*2218709) = 7*316961, etc.; given a strengthened form of the Goldbach conjecture (see Booker, 2018), one can continue iterating s^(-1) indefinitely.
Although it seems unlikely, I don't see any way to be completely certain that the forward aliquot sequence doesn't meet the backwards tree; if it did, 564 would be part of a (very long) aliquot cycle.
Many other numbers below 79750 are in a similar situation (although not 276, because it is not in the image of s).
(Added Sep 18 2021) The smallest uncertain number is 564.  All smaller numbers either have known aliquot sequences (all except 276, 306, 396, and 552), are not in the image of s (276, 306, and 552), or are in the image of s but not the image of s^2 (396).
(End)

			The smallest sociable number cycle is {12496, 14288, 15472, 14536, 14264, 12496}.

Andrew R. Booker, Finite connected components of the aliquot graph, Math. Comp. 87 (2018), 2891-2902.
Steven Finch, Amicable Pairs and Aliquot Sequences, 2013. [Cached copy, with permission of the author]
David Moews, Perfect, amicable and sociable numbers
D. 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.
Wikipedia, Sociable number

Cf. A003416 (smallest member of each cycle), A063990 (amicable numbers), A052470.

Edited (including adding comments from David Moews that this is only conjectural) by N. J. A. Sloane, Sep 17 2021

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A090615 Smallest member of sociable quadruples.

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

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A122726 Conjectured list of sociable numbers.

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