A306596 Trajectory of n under the Reverse and Add! operation carried out in base 8 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

1021, 3623, 4327, 4382, 4404, 4413, 4444, 4500, 4502, 4518, 4522, 4528, 4530, 4575, 4592, 4594, 5117, 5502, 5629, 6270, 7806, 8247, 8607, 12503, 12527, 12535, 16319, 16383, 16815, 20711, 20975, 24751, 25015, 28351, 28415, 28671, 28775, 28791, 33757, 33766, 34254, 34286, 34757, 34781, 35268, 35276

Offset: 1

Author

A.H.M. Smeets, Feb 27 2019

Keywords

• nonn
• base

Comments

A number is considered here (presumably) a Lychrel number in base 8 if it does not reach a palindrome within 100 steps more than the actual record. For those record numbers of steps, see A306600; for the corresponding record-setting numbers, see A306599. Futhermore, a Lychrel number is considered not to reach the trajectory of any smaller Lychrel number if it does not reach a trajectory of a smaller Lychrel number within 100 steps more than the actual record. For those record number of steps see A306851, and its corresponding record setting numbers, see A306850.

For a(11) = 4522 we obtain a cyclic structure of the terms in its trajectory (starting at the 12th term in the trajectory) which can be represented by the context-free grammar with alphabet = {0,1,2,3,4,5,6,7} and production rules:

S -> S_a | S_b | S_c | S_d | S_e | S_f | S_g | S_h,

S_a -> 10 T_a 00, T_a -> 7 T_a 0 | 777670,

S_b -> 11 T_b 01, T_b -> 0 T_b 7 | 076667,

S_c -> 22 T_c 12, T_c -> 0 T_c 7 | 065557,

S_d -> 44 T_d 34, T_d -> 0 T_d 7 | 043337,

S_e -> 10 T_e 000, T_e -> 7 T_e 0 | 777670,

S_f -> 11 T_f 701, T_f -> 0 T_f 7 | 007567,

S_g -> 22 T_g 712, T_g -> 0 T_g 7 | 006357,

S_h -> 44 T_h 734, T_h -> 0 T_h 7 | 003737;

i.e., the cycle length is 8.

For all other terms up to and including a(649) = 527823, no such structure has been obtained.

Crossrefs

Cf. A306600, A306599, A306851, A306850.

Base-8 analog of A075252 (base 2), A077405 (base 3), A075421 (base 4) and A063048 (base 10).