OFFSET
1,3
COMMENTS
Also called "stopping times", although that term is usually reserved for A006666.
From K. Spage, Oct 22 2009, corrected Aug 21 2014: (Start)
Congruency relationship: For n>1 and m>1, all m congruent to n mod 2^(a(n)) have a dropping time equal to a(n).
By refining the definition of the dropping time to "starting with x=n, iterate x until (abs(x) <= abs(n))" the above congruency relationship holds for all nonnegative values of n and all positive or negative values of m including zero.
By this refined definition, a(1)=2 rather than the usual zero set by convention. All other values of positive a(n) remain unchanged. (End)
Terras defines a coefficient stopping time (definition 1.5) tau(n) = d which is the smallest d for which 3^u/2^d < 1 where u is the number of tripling steps among the first d steps starting from n. Clearly tau(n) <= a(n), and Terras conjectures (conjecture 2.9) that tau(n) = a(n) for n>=2. - Olivier Rozier, May 13 2024
REFERENCES
J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010. See p. 33.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
J. C. Lagarias, The 3x+1 Problem: An Annotated Bibliography (1963-2000)., arXiv:math/0309224 [math.NT], (cit. 2007/03/08).
R. Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976) 241-252, with definition 0.1 chi(n) = a(n).
EXAMPLE
s(15) = 7, since the trajectory {T^(k)(15)} (k=1,2,3,...) equals 23,35,53,80,40,20,10.
MATHEMATICA
Collatz2[n_] := If[n<2, {}, Rest[NestWhileList[If[EvenQ[#], #/2, (3 # + 1)/2] &, n, # >= n &]]]; Table[Length[Collatz2[n]], {n, 1, 1000}]
CROSSREFS
See A074473, which is the main entry for dropping times.
KEYWORD
nonn
AUTHOR
Christof Menzel (christof.menzel(AT)hs-niederrhein.de), Mar 08 2007
EXTENSIONS
Broken link fixed by K. Spage, Oct 22 2009
STATUS
approved