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A288493 First differences of A006878 (record new trajectory lengths of Collatz function) (Hailstone sequence).

%I A288493 #46 Aug 06 2024 09:55:35
%S A288493 1,6,1,8,3,1,3,88,1,3,3,3,3,3,3,13,1,26,8,3,1,26,8,21,24,6,8,3,3,26,3,
%T A288493 13,16,11,3,21,8,3,57,6,21,39,16,3,3,26,3,3,21,13,16,52,21,3,3,13,1,
%U A288493 39,205,1,3,3,8,1,21,1,13,8,42,37,44,1,21,31,26,3,6,1,8,6,8,13,52,1,13,3,8,3,13,8,52,3,26,3,3,106,1,13,3,3,16,3,13,16,21,13,8
%N A288493 First differences of A006878 (record new trajectory lengths of Collatz function) (Hailstone sequence).
%C A288493 The sequence appears to return to 1 again and again forever when the minimal possible new record of just (previous record + 1) is reached at the latest possible value of 2X. Through 129 there are only 2 entries, a(17) and a(21), that are a minimal new record but aren't 2X.
%H A288493 Hugo Pfoertner, <a href="/A288493/b288493.txt">Table of n, a(n) for n = 1..147</a> (from Eric Rosendaal's 3x+1 Delay Records, terms 1..129 from David Rabahy)
%H A288493 Eric Roosendaal, <a href="http://www.ericr.nl/wondrous/delrecs.html">3x+1 Delay Records</a>
%e A288493 For n = 3 the difference between A006878(4) = 8 and A006878(3) = 7 is 1.
%t A288493 (* This script is not suitable to compute a large number of terms. *)
%t A288493 terms = 40; steps[x0_] := steps[x0] = Block[{x = x0, nos = 0}, While[x != 1, If[Mod[x, 2] == 0, x = x/2, x = 3*x + 1]; nos++]; nos]; b[1] = 1; b[n_] := b[n] = Block[{x = b[n - 1] + 1}, record = steps[x - 1]; While[steps[x] <= record, x++]; x];
%t A288493 A006877 = Table[Print[b[n]]; b[n], {n, 1, terms+1}];
%t A288493 A006878 = steps /@ A006877;
%t A288493 Differences[A006878] (* _Jean-François Alcover_, Jun 15 2017 *)
%Y A288493 Cf. A006877, A006878, A375094.
%K A288493 nonn
%O A288493 1,2
%A A288493 _David Rabahy_, Jun 13 2017