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Numbers whose Collatz trajectories cross their initial values a record number of times.
0

%I #13 Sep 28 2018 23:48:02

%S 1,3,6,9,14,18,33,54,97,129,194,257,294,313,342,353,398,417,470,626,

%T 9225,13739,14473,19297,27681,38881,112782,283261,427762,506977,

%U 831527,876011,1108702,1168014,1848673,2191019,5524041,8546945,10817230,10964814,11395926,13506283,15194569,41698569,46910891,52774753

%N Numbers whose Collatz trajectories cross their initial values a record number of times.

%C Indices at which records occur in A304030.

%C Number of crossings of a(n): 0, 1, 4, 5, 8, 10, 13, 14, 19, 27, 28, 29, 30, 31, 32, 37, 38, 45, 46, 50, 51, 55, 57, 59, 61, 69, 70, 71, 72, 77, 79, 81, 82, 88, 93, 97, 99, 113, 116, 120, 128, 129, 131, 139, 141, 143, ..., .

%C Also a(n) is a term of A301937.

%e 9 is a term since A304030(9) = 5 and 5 is greater than any previous term in A304030.

%t Collatz[n_] := NestWhileList[ If[ OddQ@#, 3# +1, #/2] &, n, # > 1 &]; f[n_] := Block[{x = Length[ SplitBy[ Collatz@n, # < n + 1 &]] - 1}, If[ OddQ@ n && n > 1, x - 1, x]]; mx = -1; k = 1; lst = {}; While[k < 5000001, If[ f@ k > mx, mx = f@ k; AppendTo[lst, k]]; k++]; lst

%Y Cf. A301937, A304030.

%K nonn

%O 1,2

%A _Robert G. Wilson v_, Sep 26 2018