This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A122726 #30 Feb 16 2025 08:33:02
%S A122726 12496,14264,14288,14316,14536,15472,17716,19116,19916,22744,22976,
%T A122726 31704,45946,47616,48976,83328,97946,122410,152990,177792,243760,
%U A122726 274924,275444,285778,294896,295488,358336,366556,376736,381028,418904,589786
%N A122726 Conjectured list of sociable numbers.
%C A122726 Comments from David Moews, Sep 17 2021: (Start)
%C A122726 It is possible that there are quite small numbers missing from this sequence. There is no proof that 564 (for example) is missing.
%C A122726 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))), ...
%C A122726 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.
%C A122726 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.
%C A122726 Many other numbers below 79750 are in a similar situation (although not 276, because it is not in the image of s).
%C A122726 (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).
%C A122726 (End)
%H A122726 Andrew R. Booker, <a href="https://doi.org/10.1090/mcom/3299">Finite connected components of the aliquot graph</a>, Math. Comp. 87 (2018), 2891-2902.
%H A122726 Steven Finch, <a href="/A000396/a000396.pdf">Amicable Pairs and Aliquot Sequences</a>, 2013. [Cached copy, with permission of the author]
%H A122726 David Moews, <a href="http://djm.cc/amicable.html">Perfect, amicable and sociable numbers</a>
%H A122726 D. Moews, <a href="http://djm.cc/sociable.txt">A list of currently known aliquot cycles of length greater than 2</a> [This list is not known to be complete.]
%H A122726 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SociableNumbers.html">Sociable Numbers</a>.
%H A122726 Wikipedia, <a href="http://en.wikipedia.org/wiki/Sociable_number">Sociable number</a>
%e A122726 The smallest sociable number cycle is {12496, 14288, 15472, 14536, 14264, 12496}.
%Y A122726 Cf. A003416 (smallest member of each cycle), A063990 (amicable numbers), A052470.
%K A122726 nonn
%O A122726 1,1
%A A122726 _Tanya Khovanova_, Sep 23 2006
%E A122726 Edited (including adding comments from David Moews that this is only conjectural) by _N. J. A. Sloane_, Sep 17 2021