A339769 Number of unique heights that are achieved by only one starting number in the Collatz (or '3x+1') problem when starting from numbers in the range [2^n, 2^(n+1)).

1, 2, 4, 4, 5, 6, 10, 12, 9, 7, 4, 9, 13, 11, 7, 6, 8, 10, 13, 14, 10, 15, 14, 21, 8, 7, 13, 21

Offset

0,2

Comments

The number of unique heights in each interval [2^n, 2^(n+1)) appears to be linear in n - with large oscillations about the best fit line - while the values of attained maximum heights appear to be quadratic in n.

Neither this sequence nor the corresponding sequence of maximum heights in each interval is monotone, e.g.: a(23..27) = (21, 8, 7, 13, 21) and the corresponding maximum heights are (704, 702, 949, 947, 956).

Links

Table of n, a(n) for n=0..27.

Example

a(5)=6 since the 6 heights 5, 26, 34, 109, 29, 104 are uniquely attained from the starting numbers 32, 33, 39, 41, 43, 47, respectively. The largest of the distinct heights (A280341) in the interval [32,64) however is 112.
a(11)=9 with largest unique height 237 for starting value 3711 in interval [2^11, 2^12) also is the largest height for all starting values in the interval.

Mathematica

collatz[n_] := If[EvenQ[n], n/2, 3n+1]
height[n_] := Length[NestWhileList[collatz, n, #!=1&]] - 1
a339769[n_] := Module[{heightL={}, countL={}, s, h, p}, For[s=2^n, s<2^(n+1), s++, h=height[s]; If[!MemberQ[heightL, h], AppendTo[heightL, h]; AppendTo[countL, 1], {{p}}=Position[heightL, h]; countL[[p]]+=1]]; Length[Select[Transpose[{heightL, countL}], #[[2]]==1&]]]
(* sequence data; long computation times for n >= 22 *)
Map[a339769, Range[0, 27]]

Crossrefs

Cf. A006577, A277109, A280341.

Sequence in context: A085898 A058679 A058568 * A147534 A118001 A182414

Adjacent sequences: A339766 A339767 A339768 * A339770 A339771 A339772

Keyword

nonn,more

Author

Hartmut F. W. Hoft, Dec 16 2020

Status

approved