Odd numbers n such that evenN(c)/oddN(c) is a new record, where c is the Collatz iteration of n.
1, 3, 5, 13, 21, 85, 341, 1365, 5461, 21845, 87381, 349525, 1398101
1
The functions oddN and evenN count the number of odd and even numbers. Starting at 21, the sequence appears to grow as a(n+1) = 4*a(n) + 1. The Collatz iteration stops at 1.
T. D. Noe, Plot of 13 terms
Eric W. Weisstein, MathWorld: Collatz Problem
Wikipedia, Collatz conjecture
(Mma) Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; mx = -1; t = {}; Do[c = Collatz[n]; r = Length[Select[c, EvenQ]]/Length[Select[c, OddQ]]; If[r > mx, mx = r; Print[{N[r], n}]; AppendTo[t, n]], {n, 1, 100000, 2}]; t
Cf. A248037.
nonn,more
T. D. Noe, Jan 06 2015