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A335296 Least index a(n) such that the sequences b(n,m) from A334539 are purely periodic after a(n).

%I A335296 #18 Jun 09 2021 02:23:58
%S A335296 1,1,1,1,1,1,16,24,70,31,98,112,116,170,216,488,2012,795,328,219,2993,
%T A335296 4486,1555,814,3575,12296,18386,29659,13665,2162,47685,52346,69061,
%U A335296 447927,472933,33798,857812,179171,47447,1195784,332172,618783,248092,3947618,2718980,15924182,2857983,3536883,8606700
%N A335296 Least index a(n) such that the sequences b(n,m) from A334539 are purely periodic after a(n).
%C A335296 By the pigeonhole principle, a(n) is upper bounded by n^n - n.
%H A335296 Elad Michael, <a href="/A335296/b335296.txt">Table of n, a(n) for n = 1..100</a>
%H A335296 Reddit user supermac30, <a href="https://www.reddit.com/r/math/comments/gdsjth/foggy_sequences/">Foggy Sequences</a>.
%e A335296 The sequence b(3, m) is 1, 1, 2, 1, 2, 2, 2, 3, 1, 1, 2, ... which is periodic at index 1 with period 8.
%e A335296 The sequence b(8, m) is 1, 1, 2, 1, 3, 1, 4, ... 3, 4, 1, 2, 3, 3, 4, 2, 2, 3, 3, 4, 2, 3, 3, 4, 2, 2, 3, 3, ... which is periodic at index 24 with period 9.
%o A335296 (Python)
%o A335296 def a(n):
%o A335296     b = [1];
%o A335296     for i in range(2,n+1):
%o A335296         b.append(b.count(b[-1]));
%o A335296     prev = {tuple(b):1};
%o A335296     m = 1;
%o A335296     while(True):
%o A335296         b.append(b.count(b[-1]));
%o A335296         del b[0];
%o A335296         m += 1;
%o A335296         if(tuple(b) in prev):
%o A335296             return prev[tuple(b)]
%o A335296         else:
%o A335296             prev[tuple(b)] = m;
%Y A335296 Cf. A334539.
%K A335296 nonn
%O A335296 1,7
%A A335296 _Elad Michael_, May 30 2020