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

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, 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, 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

Offset

1,2

Comments

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.

Links

Hugo Pfoertner, Table of n, a(n) for n = 1..147 (from Eric Rosendaal's 3x+1 Delay Records, terms 1..129 from David Rabahy)

Eric Roosendaal, 3x+1 Delay Records

Example

For n = 3 the difference between A006878(4) = 8 and A006878(3) = 7 is 1.

Mathematica

(* This script is not suitable to compute a large number of terms. *)
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];
A006877 = Table[Print[b[n]]; b[n], {n, 1, terms+1}];
A006878 = steps /@ A006877;
Differences[A006878] (* Jean-François Alcover, Jun 15 2017 *)

Crossrefs

Cf. A006877, A006878, A375094.

Sequence in context: A258404 A244692 A248589 * A195293 A230763 A371938

Adjacent sequences: A288490 A288491 A288492 * A288494 A288495 A288496

Keyword

nonn

Author

David Rabahy, Jun 13 2017

Status

approved