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A340739
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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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1
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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, 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, 5, 7, 4, 3, 2, 3
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OFFSET
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1,1
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COMMENTS
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The sequence is column c in the link funct.pdf with obvious adjustments.
"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:
For x a positive integer. f(x) := 3x + 1 with all positive powers of 2 remove.Note 1 is a fixed point of f.
The range of the 3x+1 function is two disjoint sets 6N+1 and 6N+5 for N nonnegative integers.
See the link to paper ammprob-f4-1-2, for a proof for range of the 3x+1 function.
Observations, Conjectures:
The famous 3x+1 problem would be solved if and only if ALL stopping time values are finite.
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}
a(n)=5 iff Mod_{128}(n) is in {13,29,33,49,63,79,101,16,56,72,76,79,92,106,122}
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}
Pattern seems to be a(n)=c iff there exist k and sets A,B such that
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.
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.
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REFERENCES
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Ultimate Challenge: the 3x+1 problem, J.C. Lagarias - editor, AMS 2010.
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LINKS
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FORMULA
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For n a positive integer,
a(n) = Min_{e=1,2,...} f^e(x(n)) < x(n), where f is the 3x+1 function defined above and
x(n) = 6n+1 if n=1,3,5,.. (odd) and x(n) = 6n-1 if n=2,4,6,... (even).
See original stopping time file, funct, before adjustments.
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MAPLE
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See links cutoff.mw.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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