A131884 Numbers conjectured to have an infinite, aperiodic, aliquot sequence.

276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098, 1104, 1134, 1218, 1302, 1314, 1320, 1338, 1350, 1356, 1392, 1398, 1410, 1464, 1476, 1488, 1512, 1560, 1572, 1578, 1590, 1632, 1650, 1662, 1674, 1722, 1734, 1758, 1770, 1806, 1836

Offset: 1

Author

J. Lowell, Oct 24 2007

Keywords

• hard
• nonn

Comments

From Martin Renner, Oct 28 2011: (Start)

There are 12 numbers up to 1000 with the five yet unknown trajectories

(1) 276 ->

306 -> 396 -> 696 -> ...

(2) 552 -> 888 -> ...

(3) 564 -> 780 -> ...

(4) 660 ->

828 ->

996 -> 1356 -> ...

(5) 966 -> 1338 -> ...

The least starting numbers 276, 552, 564, 660 and 966 for the trajectories are called Lehmer five.

There are currently 81 open end trajectories up to 10000. (End)

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

Links

• Christophe Clavier: Aliquot sequences (with leading term < 10,000).
• Wolfgang Creyaufmüller: Primzahlfamilien - aliquot sequences.
• Paul Zimmermann: Aliquot sequences.

Crossrefs

Cf. A098007, A216072, A008892, A115350.

Programs

Mathematica

(* 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 *)

Extensions

More terms and links from Martin Renner, Oct 28 2011