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A368878 a(n) is the least k such that A368877^k(n) < n or -1 if no such k exists.

%I A368878 #20 Sep 30 2024 15:47:45
%S A368878 2,1,1,2,3,1,2,1,2,1,1,2,2,1,1,1,1,1,1,1,1,1,1,1,8,1,1,2,8,1,2,1,1,1,
%T A368878 1,1,2,1,8,1,2,1,1,1,8,1,1,1,1,1,1,7,7,1,2,1,2,2,2,8,8,1,1,1,1,1,1,1,
%U A368878 6,1,1,1,1,1,1,2,2,1,1,7,7,1,1,1,1,1,1,1,6,1
%N A368878 a(n) is the least k such that A368877^k(n) < n or -1 if no such k exists.
%C A368878 This is the falling time function ft in the paper of Eliahou et al.
%C A368878 The offset is 3 because A368877(1) = A368877(2) = 2, so for n<3 is not defined.
%H A368878 Paolo Xausa, <a href="/A368878/b368878.txt">Table of n, a(n) for n = 3..10000</a>
%H A368878 Shalom Eliahou, Jean Fromentin, and Rénald Simonetto, <a href="https://hal.science/hal-03294829">Is the Syracuse falling time bounded by 12?</a>, hal-03294829, 2021.
%t A368878 A368877[n_] := Nest[If[OddQ[#], (3*#+1)/2, #/2] &, n, BitLength[n]];
%t A368878 A368878[n_] := Length[NestWhileList[A368877, n, #>=n&]]-1;
%t A368878 Array[A368878, 120, 3] (* _Paolo Xausa_, Jan 08 2024 *)
%o A368878 (PARI) T(n) = if (n%2, (3*n+1)/2, n/2); \\ A014682
%o A368878 jp(n) = my(N=1+logint(n, 2)); for (i=1, N, n = T(n)); n; \\ A368877
%o A368878 a(n) = my(k=1, m=n); while ((m=jp(m)) >= n, k++); k;
%Y A368878 Cf. A014682, A070939, A368877.
%K A368878 nonn
%O A368878 3,1
%A A368878 _Michel Marcus_, Jan 08 2024