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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)).

%I A335569 #22 Feb 13 2021 14:21:50
%S A335569 0,7,16,19,111,112,118,127,143,178,181,237,261,275,307,339,353,442,
%T A335569 469,524,556,596,664,704,705,949,950,956,964,986,1008,1050,1131,1210,
%U A335569 1219,1220,1234,1307,1321
%N 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)).
%C A335569 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 A335569 max height      start values          previous max height
%C A335569 a(5) = 112      54, 55                a(4) = 111
%C A335569 a(7) = 127      231, 235              a(6) = 118
%C A335569 a(24)= 705      31466382, 31466383    a(23)= 704
%C A335569 a(26)= 950      127456254, 127456255  a(25)= 949
%C A335569 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 A335569 a(35) = 1220 is the smallest term having 3 start values achieving maximum height: 63389366646, 63389366647, 64375365601. - _Bert Dobbelaere_, Feb 13 2021
%t A335569 collatz[n_] := If[EvenQ[n], n/2, 3n+1]
%t A335569 height[n_] := Length[NestWhileList[collatz, n, #!=1&]] - 1
%t A335569 a335569[n_] := Max[Map[height, Range[2^n, 2^(n+1)-1]]]
%t A335569 (* sequence data; long computation times for n >= 22 *)
%t A335569 Map[a335569, Range[0, 27]]
%Y A335569 Cf. A006577, A280341, A339769.
%K A335569 nonn,more
%O A335569 0,2
%A A335569 _Hartmut F. W. Hoft_, Jan 26 2021
%E A335569 a(28)-a(38) from _Bert Dobbelaere_, Feb 13 2021