|
| |
|
|
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.
|
|
14
|
|
|
|
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).
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 * A019777 A090885 A008476
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
|
| |
|
|
