A335296 - OEIS

%I #18 Jun 09 2021 02:23:58

%S 1,1,1,1,1,1,16,24,70,31,98,112,116,170,216,488,2012,795,328,219,2993,

%T 4486,1555,814,3575,12296,18386,29659,13665,2162,47685,52346,69061,

%U 447927,472933,33798,857812,179171,47447,1195784,332172,618783,248092,3947618,2718980,15924182,2857983,3536883,8606700

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

%C By the pigeonhole principle, a(n) is upper bounded by n^n - n.

%H Elad Michael, <a href="/A335296/b335296.txt">Table of n, a(n) for n = 1..100</a>

%H Reddit user supermac30, <a href="https://www.reddit.com/r/math/comments/gdsjth/foggy_sequences/">Foggy Sequences</a>.

%e 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 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 (Python)

%o def a(n):

%o     b = [1];

%o     for i in range(2,n+1):

%o         b.append(b.count(b[-1]));

%o     prev = {tuple(b):1};

%o     m = 1;

%o     while(True):

%o         b.append(b.count(b[-1]));

%o         del b[0];

%o         m += 1;

%o         if(tuple(b) in prev):

%o             return prev[tuple(b)]

%o         else:

%o             prev[tuple(b)] = m;

%Y Cf. A334539.

%K nonn

%O 1,7

%A _Elad Michael_, May 30 2020