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 A216200 #35 Nov 22 2019 02:33:38
%S A216200 1,2,2,2,3,3,3,3,4,5,6,5,5,5,5,6,7,6,7,7,8,9,10,8,9,10,11,11,12,12,11,
%T A216200 10,11,12,13,13,14,14,14,14,15,13,14,14,15,16,17,15,16,17,18,19,20,19,
%U A216200 20,19,19,20,21,19,20,20,20,21,22,23,24,24,25,26,27
%N A216200 Number of disjoint trees that appear while iterating the sum of divisors function up to n.
%C A216200 A tree like (2, 3, 4, 7, 8) contains all numbers below 8 such that iterating the sum of divisors function to any of them, while staying below 8, will lead to 8.
%C A216200 Inspired by the article in link, where a (p1, p2, p3)-tree is defined with p1 the smallest number in the tree, and p2, p3, such that all sequences {sigma^i(n)} (iterations of sigma), with p1 <= n <= p2 and sigma^i(n) < p3 have nonempty intersection with {sigma^i(p1)}. For instance, 21 (p1, 200, 10^10)-trees and 64 (p1, 1000, 10^100)-trees were found.
%H A216200 M. F. Hasler, <a href="/A216200/b216200.txt">Table of n, a(n) for n = 1..10000</a>
%H A216200 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. 93-100.
%F A216200 For n > 1, a(n) = a(n-1) + 1 - A054973(n), a(1) = 1. - _Michel Marcus_, Oct 22 2013
%F A216200 It appears that a(n)/n = 0.32721... + O(1/sqrt(n)). - M. F. Hasler, Nov 19 2019
%e A216200 For n=23, there are 10 disjoint trees: (1), (2, 3, 4, 7, 8, 15), (5, 6, 11, 12), (9, 13, 14), (10, 17, 18), (16), (19, 20), (21), (22), (23). With the arrival of 24, 3 trees are united, that is those that contain 15, 14 and 23, so that there are now 8 trees.
%e A216200 Some further values: a(100) = 33, a(500) = 167, a(1000) = 333.
%e A216200 Further values: a(10^4) = 3282, a(10^5) = 32739, a(10^6) = 327165, a(10^7) = 3272134. - _M. F. Hasler_, Nov 19 2019
%o A216200 From _M. F. Hasler_, Nov 19 2019: (Start)
%o A216200 (PARI)
%o A216200 A216200_vec(N)={my(C=Map(), s, c); vector(N, n, mapisdefined(C,n)&& c+=mapget(C,n) + mapdelete(C,n); mapput(C, s=sigma(n), if(mapisdefined(C,s), mapget(C,s))+1); n-c)} \\ Use allocatemem() for N >= 10^6.
%o A216200 A216200(n)={my(C=Map(), s); n-sum(n=2,n, mapput(C, s=sigma(n), if(mapisdefined(C,s), mapget(C,s))+1); if(mapisdefined(C,n), mapget(C,n) + mapdelete(C,n)))} \\ (slightly faster to compute a single value)
%o A216200 tree(n)=[n,if(n>1,apply(self,invsigma(n)),"fixed point")] \\ to create the tree rooted in n. (End)
%Y A216200 Cf. A000203.
%Y A216200 Cf. A257669, A257670: size and smallest number of subtree rooted at n.
%K A216200 nonn
%O A216200 1,2
%A A216200 _Michel Marcus_, Mar 12 2013