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A334245 Let S = A061228, T = A070229, and F = S o T (where o denotes composition). Then a(n) is the least number in the intersection of the sequences F^k(n) and F^k(S(n)), k>=0, or the intersection of the sequences T(F^k(n)) and T(F^k(S(n))), k>=0, or 0 if these intersections are empty.

%I A334245 #37 Dec 23 2024 14:53:46
%S A334245 12,12,12,15,12,30,15,21,15,77,21,77,30,77,30,77,30,57,77,30,77,91,77,
%T A334245 77,77,77,35,105,77,133,77,91,77,60,77,111,57,77,77,133,77,165,91,65,
%U A334245 91,147,60,105,65,70,65,165,77,105,105,91,105,180,133,231,133,91,105,105,105,234,105,147,91
%N A334245 Let S = A061228, T = A070229, and F = S o T (where o denotes composition).  Then a(n) is the least number in the intersection of the sequences F^k(n) and F^k(S(n)), k>=0, or the intersection of the sequences T(F^k(n)) and T(F^k(S(n))), k>=0, or 0 if these intersections are empty.
%C A334245 60420 >= a(n) > 0 for n <= 6042, but either a(6043) = 0 or a(6043) > 10^30.
%C A334245 If p is prime, a(p) = a(2*p).
%C A334245 From n = 5, 55, 2, 24, 245, ... begin successive strings of exactly 1, 2, 3, 4, 5, ... identical merging points that are: 15, 105, 12, 77, 713, ... . - _Bernard Schott_, Jul 04 2020
%H A334245 Robert Israel, <a href="/A334245/b334245.txt">Table of n, a(n) for n = 2..6042</a>
%H A334245 A. Sada et al., <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2020-April/020609.html">Will this pattern continue</a>, Discussion in SeqFan mailing list, Apr 2020.
%H A334245 Bernard and Blandine Schott, <a href="/A334245/a334245.pdf">Network with terms < 100</a>.
%F A334245 a(2^m+2) = a(2^m). - _Bernard Schott_, Jun 27 2020
%e A334245 For n=5, 15 = T(F(5)) = T(S(5)) so a(5) = 15.
%e A334245 For n=6, 12 = F(6) = F(S(6)) so a(6) = 12.
%e A334245 For n=10, 15 = T(10) = T(F(10)) so a(10) = 15.
%p A334245 S:= x -> x + min(numtheory:-factorset(x)):
%p A334245 T:= x -> x + max(numtheory:-factorset(x)):
%p A334245 f:= proc(n) local s,t,i,j,ct;
%p A334245   s:= S(n); i:= 0; t:= T(n); j:= 1;
%p A334245   for ct from 1 to 1000 do
%p A334245     if s = t and i=j then return s fi;
%p A334245     if s <= t then
%p A334245       if i = 0 then s:= T(s); i:= 1
%p A334245       else s:= S(s); i:= 0
%p A334245       fi
%p A334245     elif j = 0 then t:= T(t); j:= 1
%p A334245     else t:= S(t); j:= 0
%p A334245     fi
%p A334245   od;
%p A334245   0
%p A334245 end proc:
%p A334245 map(f, [$2..100]);
%t A334245 S[x_] := x + FactorInteger[x][[1, 1]];
%t A334245 T[x_] := x + FactorInteger[x][[-1, 1]];
%t A334245 f[n_] := Module[{s = S[n], t = T[n], i = 0, j = 1, ct}, For[ct = 1, ct <= 1000, ct++, If[s == t && i == j, Return[s]]; If[s <= t, If[i == 0, s = T[s]; i = 1, s = S[s]; i = 0], If[j == 0, t = T[t]; j = 1, t = S[t]; j = 0]]]; 0];
%t A334245 f /@ Range[2, 100] (* _Jean-François Alcover_, Aug 15 2020, after Maple *)
%Y A334245 Cf. A061228, A070229, A335859.
%K A334245 nonn,look
%O A334245 2,1
%A A334245 _Ali Sada_ and _Robert Israel_, Apr 19 2020