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)<n, where T^(k) is the k-th iterate, or infinity otherwise; a(n) is called the dropping time of 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

1,3

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.

Index entries for sequences related to 3x+1 (or Collatz) problem

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