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 A280341 Number of distinct heights achieved in the Collatz (or '3x+1') problem when starting from numbers in the range [2^n,2^(n+1)). 4
 1, 2, 4, 6, 10, 16, 26, 41, 53, 64, 74, 85, 101, 118, 128, 144, 157, 174, 195, 217, 238, 261, 281, 309, 324, 342, 364, 397 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Here the height is defined to be the number of halving and tripling steps required to reach 1. Interestingly the values in this sequence grow slowly (almost linearly) indicating that the average number of starting values with the same height increases with n. Question: Is this sequence always increasing? Definition corrected by N. J. A. Sloane, Apr 09 2020. The old definition was "Number of unique heights achieved in the Collatz (or '3x+1') problem when starting from numbers in the range [2^n,2^(n+1))." This sequence a(n) as well as the sequence of maximum heights in each interval appear to increase quadratically with n.  The odd numbers in [2^n, 2^(n+1)) , 5 <= n <= 20, create all distinct heights for the interval except for height n of number 2^n, and except for height n+3 when n is odd. - Hartmut F. W. Hoft, Dec 16 2020 LINKS EXAMPLE The heights for starting values 16 to 31 are: 4, 12, 20, 20, 7, 7, 15, 15, 10, 23, 10, 111, 18, 18, 18, 106. The unique heights are: 4, 12, 20, 7, 15, 10, 23, 111, 18, 106. Hence a(4)=10. MATHEMATICA collatz[n_] := If[EvenQ[n], n/2, 3n+1] height[n_] := Length[NestWhileList[collatz, n, #!=1&]] - 1 a280341[n_] := Length[Union[Map[height, Range[2^n, 2^(n+1)-1]]]] (* sequence data; long computation times for n >= 22 *) Map[a280341, Range[0, 27]] (* Hartmut F. W. Hoft, Dec 16 2020 *) CROSSREFS Cf. A006577, A277109. Sequence in context: A017985 A327474 A028488 * A227572 A080432 A094985 Adjacent sequences:  A280338 A280339 A280340 * A280342 A280343 A280344 KEYWORD nonn AUTHOR Dmitry Kamenetsky, Jan 01 2017 EXTENSIONS a(25)-a(27) from Hartmut F. W. Hoft, Dec 16 2020 STATUS approved

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Last modified July 26 22:49 EDT 2021. Contains 346300 sequences. (Running on oeis4.)