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Search: id:A126241
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| A126241 |
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Dropping times in the 3n+1 problem (or the Collatz problem). Let f(n):=n/2 if n is even, (3n+1)/2 otherwise. Let s(n) be the smallest integer k such that f^(k)(n)<n, where f^(k) is the k-th iterate, or infinity otherwise. s(n) is called the dropping time of n. |
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3
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| 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; listen)
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OFFSET
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1,3
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COMMENT
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Contribution from K. Spage (kevspage2001(AT)yahoo.co.uk), Oct 22 2009: (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) OR (x<=1 AND x>=0))" the above congruency relationship holds for all non-negative 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)
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LINKS
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J. C. Lagarias: The 3x+1 Problem: An Annotated Bibliography (1963-2000). (cit. 2007/03/08).
Mattfind, List of links on the 3x+1 problem [From K. Spage (kevspage2001(AT)yahoo.co.uk), Oct 22 2009]
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EXAMPLE
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s(15) = 7, since the trajectory {f^(k)(15)} (k=1,2,3,...) equals 23,35,53,80,40,20,10.
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CROSSREFS
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See A074473, which is the main entry for dropping times.
Contribution from K. Spage (kevspage2001(AT)yahoo.co.uk), Oct 22 2009: (Start)
Allowable dropping times for A126231 are listed in A020914.
a(n) = ceil(A102419(n)/(1+log(2)/log(3))) (End)
Sequence in context: A109008 A074695 A069098 this_sequence A019777 A090885 A008476
Adjacent sequences: A126238 A126239 A126240 this_sequence A126242 A126243 A126244
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KEYWORD
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nonn
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AUTHOR
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Christof Menzel (christof.menzel(AT)hs-niederrhein.de), Mar 08 2007
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EXTENSIONS
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Broken link fixed by K. Spage (kevspage2001(AT)yahoo.co.uk), Oct 22 2009
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