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A257670 Minimum term in the sigma(x) -> x subtree whose root is n.

%I A257670 #70 Dec 12 2019 11:41:26
%S A257670 1,2,2,2,5,5,2,2,9,10,11,5,9,9,2,16,17,10,19,19,21,22,23,2,25,26,27,5,
%T A257670 29,29,16,16,33,34,35,22,37,37,10,27,41,19,43,43,45,46,47,33,49,50,51,
%U A257670 52,53,34,55,5,49,58,59,2,61,61,16,64,65,66,67,67,69
%N A257670 Minimum term in the sigma(x) -> x subtree whose root is n.
%H A257670 M. F. Hasler, <a href="/A257670/b257670.txt">Table of n, a(n) for n = 1..20000</a>, Nov 19 2019 (first 1000 terms from Michel Marcus)
%H A257670 M. Alekseyev, <a href="https://home.gwu.edu/~maxal/gpscripts/invphi.gp">PARI/GP Scripts for Miscellaneous Math Problems: invsigma.gp</a>, Oct. 2005
%H A257670 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.
%H A257670 R. J. Mathar, <a href="/A257670/a257670.pdf">Illustrations</a>
%F A257670 a(m) = m for m in A007369: sigma(x) = m has no solution. [Corrected by _M. F. Hasler_, Nov 19 2019]
%F A257670 a(A007497(n)) = 2; a(A051572(n)) = 5; a(A257349(n)) = 16. (These sequences being the trajectory of 2, 5 resp. 16 under iterations of sigma = A000203.)
%e A257670 We have the following trees (a <- b means sigma(a) = b):
%e A257670   2 <-- 3 <-- 4 <-- 7 <-- 8 <-- 15 <-- 24 <-- 60 <-- ...
%e A257670                     9 <-- 13 <-- 14 <-’
%e A257670   5 <-- 6 <-- 12 <-- 28 <-- 56 <-- 120 <-- ...
%e A257670         11 <-’             /
%e A257670        10 <-- 18 <-- 39 <-’
%e A257670 The number 1 has strictly speaking an arrow to itself, so it is not part of a tree. (For all n > 1, sigma(n) > n, so no other fixed point or longer "cycle" can exist.) But actually we rather consider connected components, and let a(1) = 1 as the smallest element of this connected component.
%e A257670 a(2) = 2, since there is no smaller x such that sigma(x) = 2: the subtree with root 2 is reduced to a single node: 2. Similarly, a(m) = m for all m in A007369.
%e A257670 For n=3, since sigma(2) = 3, the tree whose root is 3 has 2 nodes: 2 and 3, and the smallest one is 2, hence a(3) = 2.
%e A257670 Similarly, although 24 occurs directly first at sigma(14), it is also reached from 15 which is in turn reached, via intermediate steps, from 2. Thus, the subtree with root 24 has as 2 as smallest element, whence a(24) = 2.
%o A257670 (PARI) lista(nn) = {my(v = vector(nn)); v[1] = 1; for (i=2, nn, my(s = i); while (s <= nn, if (v[s] == 0, v[s] = i); s = sigma(s););); for (i=1, nn, if (v[i] == 0, v[i] = i);); v;} \\ _Michel Marcus_, Nov 19 2019
%o A257670 (PARI) A257670(n)=if(n>2,vecmin(concat(apply(self,invsigma(n)),n)),n) \\ See Alekseyev-link for invsigma(). - _David A. Corneth_ and _M. F. Hasler_, Nov 20 2019
%Y A257670 Cf. A000203 (sigma), A007369  (sigma(x) = n has no solution).
%Y A257670 Cf. A216200 (number of disjoint trees), A257348 (minimal node of all trees).
%Y A257670 Cf. A257669 (number of terms in current tree).
%K A257670 nonn
%O A257670 1,2
%A A257670 _Michel Marcus_, May 03 2015
%E A257670 Edited by _M. F. Hasler_, Nov 19 2019