For the Collatz (3x+1) sequence s beginning with n, the difference between n and the next smaller number in s.
1, 1, 2, 1, 1, 2, 4, 1, 2, 1, 2, 3, 1, 5, 8, 1, 1, 2, 4, 5, 2, 3, 8, 3, 6, 4, 2, 3, 7, 8, 16, 4, 8, 15, 2, 3, 4, 1, 20, 1, 10, 3, 4, 5, 6, 1, 24, 9, 6, 7, 12, 13, 1, 2, 4, 1, 6, 1, 7, 8, 1, 2, 32, 9, 8, 9, 16, 17, 17, 1, 20, 2, 18, 11, 18, 19, 2, 3, 40, 1, 2, 3
2
In the plot, the diagonal line starting at about 9000 appears to be the last such diagonal. Why?
T. D. Noe, Plot of 20000 terms
T. D. Noe, Table of 20000 terms
(Mma) Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[s = Sort[Collatz[n] - n]; ps = Position[s, 0][[1, 1]]; -s[[ps - 1]], {n, 2, 100}]
nonn
T. D. Noe, Jan 30 2017