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A335569 a(n) is the maximum height achieved in the Collatz ('3x+1') problem when starting from numbers in the range [2^n, 2^(n+1)). 1

%I

%S 0,7,16,19,111,112,118,127,143,178,181,237,261,275,307,339,353,442,

%T 469,524,556,596,664,704,705,949,950,956,964,986,1008,1050,1131,1210,

%U 1219,1220,1234,1307,1321

%N a(n) is the maximum height achieved in the Collatz ('3x+1') problem when starting from numbers in the range [2^n, 2^(n+1)).

%C This sequence is strictly increasing since the height of number 2*k is one larger than the height of k; it appears to fit a quadratic with respect to exponent n. Through n=27 the maximum values are achieved by odd starting values and most are unique heights. The non-unique exceptions are:

%C max height start values previous max height

%C a(5) = 112 54, 55 a(4) = 111

%C a(7) = 127 231, 235 a(6) = 118

%C a(24)= 705 31466382, 31466383 a(23)= 704

%C a(26)= 950 127456254, 127456255 a(25)= 949

%C Since a(5) = a(4) + 1 and a(26) = a(25) + 1, and since probably many additional such pairs exist, maximum heights cannot be used in showing that A280341 is strictly increasing.

%e a(35) = 1220 is the smallest term having 3 start values achieving maximum height: 63389366646, 63389366647, 64375365601. - _Bert Dobbelaere_, Feb 13 2021

%t collatz[n_] := If[EvenQ[n], n/2, 3n+1]

%t height[n_] := Length[NestWhileList[collatz, n, #!=1&]] - 1

%t a335569[n_] := Max[Map[height, Range[2^n, 2^(n+1)-1]]]

%t (* sequence data; long computation times for n >= 22 *)

%t Map[a335569, Range[0, 27]]

%Y Cf. A006577, A280341, A339769.

%K nonn,more

%O 0,2

%A _Hartmut F. W. Hoft_, Jan 26 2021

%E a(28)-a(38) from _Bert Dobbelaere_, Feb 13 2021

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Last modified September 20 11:42 EDT 2021. Contains 347584 sequences. (Running on oeis4.)