For odd n, the number of Collatz (3x+1) iteration terms greater than n.
0, 5, 2, 12, 13, 8, 3, 11, 4, 10, 2, 7, 10, 102, 5, 97, 10, 5, 4, 17, 97, 10, 3, 92, 5, 7, 2, 98, 9, 11, 6, 93, 5, 6, 2, 87, 97, 7, 3, 13, 6, 92, 2, 8, 6, 76, 4, 87, 94, 5, 2, 70, 13, 80, 85, 51, 4, 7, 2, 9, 75, 18, 82, 24, 89, 5, 2, 12, 70, 12, 3, 81, 82, 86, 2
1
A new maximum appears infrequently; see S001016 for the location of new maximums.
T. D. Noe, Plot of 5000 terms
T. D. Noe, Table of 5000 terms
Example: when n=3, the Collatz iteration is 3, 10, 5, 16, 8, 4, 2, 1 and it is clear that there are 5 terms greater than 3.
Wikipedia, Collatz conjecture
(Mma) Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[Length[Select[Collatz[n], # > n &]], {n, 1, 200, 2}]
Cf. A222597, S001016, S001020.
nonn
T. D. Noe, May 08 2017