%I #48 Nov 30 2019 11:35:48

%S 1,2,5,16,19,27,29,33,49,50,52,66,81,85,105,146,147,163,170,189,197,

%T 199,218,226,243,262,303,315,343,424,430,438,453,461,463,469,472,484,

%U 489,513,530,550,584,677,722,746,786,787,804,813,821,831,842,859,867,876,892,903,914,916,937,977,982,988,990,1029

%N Repeatedly applying the map x -> sigma(x) partitions the natural numbers into a number of disjoint trees; sequence gives the (conjectural) list of minimal representatives of these trees.

%C Very little is known for certain. Even the trajectories of 2 (A007497) and 5 (A051572) under repeated application of the map x -> sigma(x) (cf. A000203) are only conjectured to be disjoint.

%C The thousand-term b-file (up to 141441) has been checked to correspond to disjoint trees for 265 iterations of sigma on each term, and every non-term n < 141441 merges (in at most 21 iterations) with an earlier iteration sequence. - _Hans Havermann_, Nov 22 2019

%C Rather than trees we mean connected components of the graphs with edges x -> sigma(x). The number 1 is a fixed point, i.e., a cycle of length 1 under iterations of sigma, it is not part of a tree. But since sigma(n) > n for n > 1 there are no other cycles. - _M. F. Hasler_, Nov 21 2019

%D Kerry Mitchell, Posting to Math Fun Mailing List, Apr 30 2015

%H Hans Havermann, <a href="/A257348/b257348.txt">Table of n, a(n) for n = 1..1000</a>

%H G. L. Cohen and H. J. J. te Riele, <a href="http://projecteuclid.org/euclid.em/1047565640">Iterating the sum-of-divisors function</a>, Experimental Mathematics, 5 (1996), pp. 91-100. See Eq. (4.2).

%Y Cf. A000203 (sigma), A007497 (trajectory of 2), A051572 (trajectory of 5), A257349 (trajectory of 16).

%Y Cf. A216200 (number of disjoint trees up to n); A257669 and A257670: size and smallest number of subtree rooted in n.

%K nonn,hard

%O 1,2

%A _N. J. A. Sloane_, May 01 2015, following a suggestion from _Kerry Mitchell_.

%E More terms from _Hans Havermann_, May 02 2015