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Number of iterations of the map n -> f(f(f(...f(n)...))) to reach the end of the cycle, where f(n) = A006577(n), the initial number n is not counted.
1

%I #22 Oct 27 2019 12:00:49

%S 0,1,5,2,0,7,4,6,7,8,11,8,8,10,10,3,9,6,6,5,5,11,11,9,12,9,13,7,7,7,

%T 10,1,10,9,9,6,6,6,10,7,11,7,8,4,4,4,10,12,10,10,10,12,12,7,7,7,2,7,2,

%U 7,7,15,15,8,14,14,14,11,11,11,14,12,12,12,11,12

%N Number of iterations of the map n -> f(f(f(...f(n)...))) to reach the end of the cycle, where f(n) = A006577(n), the initial number n is not counted.

%C A006577 is the number of halving and tripling steps to reach 1 in '3x+1' problem.

%C The end of the cycle is 1 or 5 for n = 5, 32, 57, 59, 344, 346, 348, 349, ...

%H Michel Lagneau, <a href="/A213198/b213198.txt">Table of n, a(n) for n = 1..10000</a>

%e a(3) = 5 because the 5 iterations to reach 1 are A006577(3) = 7; A006577(7) = 16; A006577(16) = 4; A006577(4) = 2; A006577(2) = 1.

%e a(5) = 0 because A006577(5) = 5 is the end of the cycle.

%e a(57) = 2 because A006577(57) = 32 and A006577(32) = 5 is the end of the cycle.

%p for n from 1 to 200 do:

%p m:=n: a:=2:

%p for it from 1 to 1000

%p while (a>1) do:

%p jj:=0: a:=0: x:=m:

%p if m=5 then

%p printf(`%d, `, it-1): jj:=1:

%p else

%p for i from 1 to 1000

%p while (x>1) do:

%p if irem(x, 2)=0 then

%p x := x/2: a := a+1:

%p else

%p x := 3*x+1: a := a+1:

%p fi:

%p od:

%p m:=a:

%p fi:

%p od:

%p if jj=0 then

%p printf(`%d, `, it-1):

%p fi:

%p od:

%t Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; f[n_] := Length[Collatz[n]] - 1; Table[k = Rest[NestWhileList[f, n, UnsameQ, All]]; If[k[[1]] == n, 0, k = DeleteCases[k, 0]; If[Length[k] > 1 && k[[-1]] == k[[-2]], k = Most[k]]; Length[k]], {n, 100}] (* _T. D. Noe_, Mar 01 2013 *)

%Y Cf. A006577.

%K nonn

%O 1,3

%A _Michel Lagneau_, Mar 01 2013