%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 nonunique 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
