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

Offset: 1

Author

Christof Menzel (christof.menzel(AT)hs-niederrhein.de), Mar 08 2007

Keywords

• nonn

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

Examples

			s(15) = 7, since the trajectory {T^(k)(15)} (k=1,2,3,...) equals 23,35,53,80,40,20,10.
		

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 (1963-1999), arXiv:math/0309224 [math.NT], 2003-2011.
• Olivier Rozier and Claude Terracol, Paradoxical behavior in Collatz sequences, arXiv:2502.00948 [math.GM], 2025. See p. 2.
• Riho Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976) 241-252, with definition 0.1 chi(n) = a(n).
• Index entries for sequences related to 3x+1 (or Collatz) problem

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

Programs

Mathematica

Collatz2[n_] := If[n<2, {}, Rest[NestWhileList[If[EvenQ[#], #/2, (3 # + 1)/2] &, n, # >= n &]]]; Table[Length[Collatz2[n]], {n, 1, 1000}]

Formula

a(n) = ceiling(A102419(n)/(1+log(2)/log(3))). - K. Spage, Aug 22 2014

Extensions

Broken link fixed by K. Spage, Oct 22 2009