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%I #27 Sep 10 2015 11:24:39
%S 276,306,396,552,564,660,696,780,828,888,966,996,1074,1086,1098,1104,
%T 1134,1218,1302,1314,1320,1338,1350,1356,1392,1398,1410,1464,1476,
%U 1488,1512,1560,1572,1578,1590,1632,1650,1662,1674,1722,1734,1758,1770,1806,1836
%N Numbers conjectured to have an infinite, aperiodic, aliquot sequence.
%C From _Martin Renner_, Oct 28 2011: (Start)
%C There are 12 numbers up to 1000 with the five yet unknown trajectories
%C (1) 276 ->
%C 306 -> 396 -> 696 -> ...
%C (2) 552 -> 888 -> ...
%C (3) 564 -> 780 -> ...
%C (4) 660 ->
%C 828 ->
%C 996 -> 1356 -> ...
%C (5) 966 -> 1338 -> ...
%C The least starting numbers 276, 552, 564, 660 and 966 for the trajectories are called Lehmer five.
%C There are currently 81 open end trajectories up to 10000. (End)
%C Sequence A216072 lists only the values that are the lowest starting elements of open end aliquot sequences that are the part of different open-ending families. But this sequence lists all the starting values of an aliquot sequence that lead to open-ending. It includes all values obtained by iterating from the starting values of this sequence. - _V. Raman_, Dec 08 2012
%H Christophe Clavier: <a href="http://christophe.clavier.free.fr/Aliquot/site/Aliquot.html">Aliquot sequences (with leading term < 10,000)</a>.
%H Wolfgang Creyaufmüller: <a href="http://www.aliquot.de/aliquot.htm">Primzahlfamilien - aliquot sequences</a>.
%H Paul Zimmermann: <a href="http://www.loria.fr/~zimmerma/records/aliquot.html">Aliquot sequences</a>.
%t (* This script is not suitable for a large number of terms *) 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[n]; AppendTo[A131884, n]; False, Length[{##}] < 4 || {##}[[-4 ;; -3]] != {##}[[-2 ;; -1]]] &, All] == 1; selQ /@ Range[1000]; A131884 (* _Jean-François Alcover_, Sep 10 2015 *)
%Y Cf. A098007, A216072, A008892, A115350.
%K hard,nonn
%O 1,1
%A _J. Lowell_, Oct 24 2007
%E More terms and links from _Martin Renner_, Oct 28 2011
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