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 A126241 Dropping times in the 3n+1 problem (or the Collatz problem). Let T(n):=n/2 if n is even, (3n+1)/2 otherwise (A014682). Let a(n) be the smallest integer k such that T^(k)(n)
 0, 1, 4, 1, 2, 1, 7, 1, 2, 1, 5, 1, 2, 1, 7, 1, 2, 1, 4, 1, 2, 1, 5, 1, 2, 1, 59, 1, 2, 1, 56, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 5, 1, 2, 1, 54, 1, 2, 1, 4, 1, 2, 1, 5, 1, 2, 1, 7, 1, 2, 1, 54, 1, 2, 1, 4, 1, 2, 1, 51, 1, 2, 1, 5, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 5, 1, 2, 1, 45, 1, 2, 1, 8, 1, 2, 1, 4 (list; graph; refs; listen; history; text; internal format)
 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) 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. FORMULA a(n) = ceiling(A102419(n)/(1+log(2)/log(3))). - K. Spage, Aug 22 2014 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. Cf. A014682, A006666, A006577. Records: A060412, A060413. Cf. A020914 (allowable dropping times). - K. Spage, Aug 22 2014 Sequence in context: A187025 A074695 A069098 * A353515 A019777 A337515 Adjacent sequences:  A126238 A126239 A126240 * A126242 A126243 A126244 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

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Last modified May 18 22:00 EDT 2022. Contains 353825 sequences. (Running on oeis4.)