The record values of the length of the 3x-1 iteration.
1, 2, 4, 5, 8, 11, 15, 18, 21, 26, 28, 63, 64, 72, 73, 81, 83, 85, 101, 106, 118, 120, 143, 151, 153, 164, 187, 192, 196, 198, 204, 206, 209, 211, 213, 215, 231, 236, 240, 261, 266, 352, 356, 358, 359, 361, 377, 393, 395, 435, 436, 449, 454, 455, 463, 471, 483, 509, 511, 526, 530, 611, 613
1
See S000241 for the location of these maxima.
T. D. Noe, Plot of 63 terms
(Mma) t = {{1, 1}}; mx = 1; Do[s = {n}; m = n; While[If[EvenQ[m], m = m/2, m = (3*m - 1)/2]; ! MemberQ[s, m], AppendTo[s, m]]; If[Length[s] > mx, mx = Length[s]; AppendTo[t, {n, mx}]], {n, 2, 1000000}]; Transpose[t][[2]]
nonn
T. D. Noe, Sep 04 2014