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The analog of A065198 in base 2. Integers like 22, for which a palindrome is never reached (cf. A066059), are of course disregarded. A066145 gives the corresponding records.

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.

It is conjectured that if a Reverse and Add! trajectory reaches a palindrome, it will be reached in relatively few steps, or otherwise it will never reach a palindrome.

For all n, all numbers m < A306599(n) will reach a palindrome within a(n) Reverse and Add! steps, or otherwise it is conjectured never to reach a palindrome.