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The stopping time sequence for the 3x+1 function, restricted to its range and adjusted. Each term is the number of iterations of the function until it decreases.
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%I #26 Mar 17 2021 15:56:32

%S 4,3,2,3,35,2,3,34,3,4,2,32,5,2,28,5,26,3,2,3,9,2,3,4,3,25,2,18,5,2,4,

%T 8,5,3,2,3,19,2,3,12,3,17,2,4,4,2,15,6,5,3,2,3,13,2,3,5,3,6,2,10,6,2,

%U 5,7,4,3,2,3

%N The stopping time sequence for the 3x+1 function, restricted to its range and adjusted. Each term is the number of iterations of the function until it decreases.

%C The sequence is column c in the link funct.pdf with obvious adjustments.

%C "Adjusted" means that, since for every four terms the first two are 1's they are omitted and then the sequence is relabeled. The 3x+1 function is defined:

%C For x a positive integer. f(x) := 3x + 1 with all positive powers of 2 remove.Note 1 is a fixed point of f.

%C The range of the 3x+1 function is two disjoint sets 6N+1 and 6N+5 for N nonnegative integers.

%C See the link to paper ammprob-f4-1-2, for a proof for range of the 3x+1 function.

%C Observations, Conjectures:

%C The famous 3x+1 problem would be solved if and only if ALL stopping time values are finite.

%C a(n)=2 iff Mod_{8}(n) is in {3, 6} a(n)=3 iff Mod_{16}(n) is in {7,9, 2,4} a(n)=4 iff Mod_{64}(n) is in {1,31,45,10,24,44}

%C a(n)=5 iff Mod_{128}(n) is in {13,29,33,49,63,79,101,16,56,72,76,79,92,106,122}

%C a(n)=6 iff Mod_{512}(n) is in {61,97,241,255,293,333,337,389,399,437,477,495,48,58,96,136,154,232,268,412,426,464,504,508}

%C Pattern seems to be a(n)=c iff there exist k and sets A,B such that

%C Mod_{2^k}(n) is in A union B, where |A|=|B| and A are odd and B are even numbers, where A is associated with 6N+1 and B with 6N+5.

%C Conjecture: Ultimately every positive integer appears in the stopping time sequence. (verified up to 100, examples: a(6802394)=160, a(31229269)=161) And each positive integer is in the sequence an infinite number of times.

%D Ultimate Challenge: the 3x+1 problem, J.C. Lagarias - editor, AMS 2010.

%H Sam E. Speed, <a href="/A340739/b340739.txt">Table of n, a(n) for n = 1..8194</a>

%H Sam E. Speed, <a href="/A340739/a340739.tex">ammprob-f4-1-2.tex, proof of range of 3x+1 function</a>

%H Sam E. Speed, <a href="/A340739/a340739.mw.txt">Maple 12 program cutoff.mw used to make b-file</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Collatz_conjecture">Collatz conjecture</a>

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

%F For n a positive integer,

%F a(n) = Min_{e=1,2,...} f^e(x(n)) < x(n), where f is the 3x+1 function defined above and

%F x(n) = 6n+1 if n=1,3,5,.. (odd) and x(n) = 6n-1 if n=2,4,6,... (even).

%F See original stopping time file, funct, before adjustments.

%p See links cutoff.mw.

%Y Cf. A067745.

%K nonn

%O 1,1

%A _Sam E. Speed_, Jan 18 2021