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Number of different initial values for 3x+1 trajectories in which the largest term appearing in the iteration is 2^(6n+4).
4

%I #13 Jul 23 2024 01:01:15

%S 6,12,8,6,13,8,6,9,11,6,21,8,6,78,8,6,9,13,6,15,8,6,16,8,6,9,20,6,12,

%T 8,6,13,8,6,9,11,6,14,8,6,32,8,6,9,32,6,23,8,6,24,8,6,9,14,6,12,8,6,

%U 13,8,6,9,11,6,14,8,6,19,8,6,9,13,6,80,8,6,29,8,6,9,18,6,12,8,6,13,8,6,9,11

%N Number of different initial values for 3x+1 trajectories in which the largest term appearing in the iteration is 2^(6n+4).

%C From _Hartmut F. W. Hoft_, Jun 24 2016: (Start)

%C The sequence has the quasiperiod 6, x, 8, 6, y, 8, 6, 9, z of length 9 starting at index 0 where x, y, z > 10; in addition, a(3*9*n+1) = 12, a(3*9*n+4) = 13 and a(3*9*n+8) = 11 for all n>=0; proof by induction (see this link) as in the link in A087256 based on the modular identities in the link in A033496.

%C Conjecture: All numbers greater than 10 appear in the sequence (see also A033496 and A233293). (End)

%H Hartmut F. W. Hoft, <a href="/A105730/a105730.pdf">identities to prove for quasi period 9</a>

%F a(n) = A087256(6n+4).

%e a(1) = 12, i.e. the number of initial values for 2^10, since 804 -> 402 -> 201 -> 604 -> 302 -> 151 -> 454 -> 227 -> 682 -> 341 -> 1024 and 908 -> (454 -> ... -> 1024) are the two maximal trajectories containing all 12 initial values. a(8) = 11 since 2^(6*8+4) has 11 different initial values for Collatz trajectories leading to it. - _Hartmut F. W. Hoft_, Jun 24 2016

%t trajectory[start_] := NestWhileList[If[OddQ[#], 3#+1, #/2]&, start, #!=1&]

%t fanSize[max_] := Module[{active={max}, fan={}, current}, While[active!={}, current=First[active];active=Rest[active]; AppendTo[fan, current]; If[2*current<=max, AppendTo[active, 2*current]]; If[Mod[current, 3]==1 && OddQ[(current-1)/3] && current>4, AppendTo[active, (current-1)/3]]]; Length[fan]]/;max==Max[trajectory[max]]

%t a105730[low_, high_] := Map[fanSize[2^(6#+4)]&, Range[low, high]]

%t a105730[0,89] (* _Hartmut F. W. Hoft_, Jun 24 2016 *)

%Y Cf. A025586, A033496, A087256, A233293.

%K easy,nonn,less

%O 0,1

%A _David Wasserman_, Apr 18 2005