A339769 -id:A339769 - cp's OEIS frontend

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)).

0, 7, 16, 19, 111, 112, 118, 127, 143, 178, 181, 237, 261, 275, 307, 339, 353, 442, 469, 524, 556, 596, 664, 704, 705, 949, 950, 956, 964, 986, 1008, 1050, 1131, 1210, 1219, 1220, 1234, 1307, 1321

Offset: 0

Hartmut F. W. Hoft, Jan 26 2021

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:
max height      start values          previous max height
a(5) = 112      54, 55                a(4) = 111
a(7) = 127      231, 235              a(6) = 118
a(24)= 705      31466382, 31466383    a(23)= 704
a(26)= 950      127456254, 127456255  a(25)= 949
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.

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

Cf. A006577, A280341, A339769.

collatz[n_] := If[EvenQ[n], n/2, 3n+1]
height[n_] := Length[NestWhileList[collatz, n, #!=1&]] - 1
a335569[n_] := Max[Map[height, Range[2^n, 2^(n+1)-1]]]
(* sequence data; long computation times for n >= 22 *)
Map[a335569, Range[0, 27]]

a(28)-a(38) from Bert Dobbelaere, Feb 13 2021

cp's OEIS Frontend

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)).

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