2^n - S000272(n).
0, 3, 0, 13, 9, 0, 69, 37, 0, 25, 0, 33, 89, 0, 165, 97, 0, 65, 0, 609, 273, 0, 1177, 785, 0, 1537, 1025, 0, 1297, 0, 865, 577, 0, 769, 385, 0, 257, 0, 3041, 229, 0, 305, 153, 0, 721, 481, 0, 641, 0, 321, 4097, 0, 2533, 1689, 0, 9281, 0, 12817, 8545, 0, 6929
1
T. D. Noe, Plot of 200 terms
T. D. Noe, Table of 200 terms
(Mma) Collatz2[n_] := Most[NestWhileList[If[EvenQ[#], #/2, (3 # + 1)/2] &, n, # >= n &]]; gg[n_] := Floor[1 + n/(1 - (Log[2]/Log[3]))]; nn = 100; q = Array[gg, nn, 0]; Join[{0, 3}, Table[If[MemberQ[q, n], 0, k = 2^n - 1; While[Length[Collatz2[k]] != n, k--]; 2^n - k], {n, 3, nn}]]
Cf. S000272, S000303 (without zeros).
nonn
T. D. Noe, Oct 15 2014