cp's OEIS Frontend

For 318 (cf. A075153), 266718 (cf. A075466) and 270798 (cf. A075467) one can prove that the base 4 trajectory does not contain a palindrome. A proof for 290 (cf. A075299) has not been found up to now. 4398859679359 is another known candidate (obtained from a remark of David J. Seal, cf. Links) for a term whose trajectory is provably palindrome-free, but is not secured that it does not join the trajectory of some term m < n. - If the trajectory of an integer k joins the trajectory of a smaller integer which is a term of the present sequence, then this occurs after very few Reverse and Add! steps (at most 28 for k < 20000). On the other hand, the trajectories of the terms listed above do not join the trajectory of any smaller term within at least 1000 steps.

Base-4 analog of A063048 (base 10) and A075252 (base 2); subsequence of A075420.

From A.H.M. Smeets, Mar 18 2019: (Start)

David J. Seal (see LINKS) observed a cyclic pattern (length 6) in the trajectories that can be represented by an extended right regular grammar with production rules:

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

S_a -> 1033202000232 T_a, T_a -> 222 T_a | 2302333113230

S_b -> 2022321332331 T_b, T_b -> 111 T_b | 1223001203131

S_c -> 10002003002212 T_c, T_c -> 222 T_c | 3221333101333

S_d -> 103312202321111 T_d, T_d -> 111 T_d | 1102023122000

S_e -> 110200123122222 T_e, T_e -> 222 T_e | 2231232001301

S_f -> 213301021321111 T_f, T_f -> 111 T_f | 1113213003312

Within the first 471 terms of this sequence we observed three trajectories with a cyclic pattern (length 6) that can be represented by a context-free grammar with production rules:

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

S_a -> 10 T_a 00, T_a -> 3 T_a 0 | T_a0,

S_b -> 11 T_b 01, T_b -> 0 T_b 3 | T_b0,

S_c -> 22 T_c 12, T_c -> 0 T_c 3 | T_c0,

S_d -> 10 T_d 000, T_d -> 3 T_d 0 | T_d0,

S_e -> 11 T_e 301, T_e -> 0 T_e 3 | T_e0,

S_f -> 22 T_f 312, T_f -> 0 T_f 3 | T_f0.

The terminating strings in these context-free grammars are given by:

n 2 359 371

a(n) 318 266718 270798

T_a0 33230 33230000001033230 3323001033230

T_b0 03123 03123010001103123 0312302103123

T_c0 01313 01313120002201313 0131320201313

T_d0 33323 33323000001033323 3332300103323

T_e0 03222 03222301001103222 0322201113222

T_f0 02111 02111312002202111 0211112222111

From the fact that both, right regular grammars and context-free grammars occur, we wonder if other trajectories can be represented by context-sensitive grammars as well, by which other trajectories can be proven never to end up in a palindromic string? (End)

Conjecture 1: For each k > 0 the trajectory of k eventually leads to a term in the trajectory of some j which belongs to A063048, i.e. whose trajectory (presumably) never leads to a palindrome. Conjecture 2: There is no k > 0 such that the trajectory of k contains more than twelve palindromes, i.e. a(n) = -1 for n > 12.

Although the starting number k is regarded as part of the trajectory, it is allowed to be palindromic. Hence palindromes are not excluded from the sequence. A063048 is obtained if palindromes are excluded. The smallest term in A088753 but not in A063048 is 9999, the smallest term in A063048 but not in A088753 is 19098.

W. VanLandingham and others have computed nearly 10^7 terms (all terms < 10^14), cf. W. VanLandingham, 196 and Other Lychrel Numbers.

From M. F. Hasler, Apr 13 2019: (Start)

Lychrel numbers listed here are also called "seeds", in contrast to Kin numbers A023108 which include all terms in the orbits of the former.

It is not easy to determine whether the orbit of a given term will never merge into the orbit of an earlier term. It seems that the property of "disjoint orbit" is as conjectural as the property of not reaching a palindrome. One could specify a "search limit" in order to get a well-defined sequence. The given list of terms has been checked and extended by considering the orbits up to members of size <= 10^199 at least. Given that the number increases by a factor 10 roughly every 2.416 iterations, this corresponds to about 500 iterations. (End)

Base 3 analog of A075252 (base 2), A075421 (base 4) and A063048 (base 10); subsequence of A077404. - A proof that the base 3 trajectory does not contain a palindrome has been found up to now for none of the terms. - If the trajectory of an integer k joins the trajectory of a smaller integer which is a term of the present sequence, then this occurs after very few Reverse and Add! steps (at most 9 for k < 20000). On the other hand, the trajectories of the terms of this sequence do not join the trajectory of any smaller term within at least 1000 steps.

a(3), a(14) and a(18) are conjectural; it is not yet ensured that they are minimal.

a(n) >= A070788(n); a(n) = A070788(n) iff the trajectory of A070788(n) is palindrome-free, i.e. A070788(n) is also a term of A063048.

a(n) determines a 1-1-mapping from the terms of A070788 to the terms of A063048, the inverse of the mapping determined by A089493. Terms > 2*10^6 were ascertained with the aid of W. VanLandingham's list of Lychrel numbers.

The 1-1 property of the mapping depends on the conjecture that the Reverse and Add! trajectory of each term of A070788 contains only a finite number of palindromes (cf. A077594). - Klaus Brockhaus, Dec 09 2003

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.

For terms <= 5000 each palindrome is reached from the preceding one or from the start in at most 15 steps; after the presumably last one no further palindrome is reached in 2000 steps.

For terms < 5000000 each palindrome is reached from the preceding one or from the start in at most 35 steps; after the presumably last one no further palindrome is reached in 2000 steps.

Additional terms are 20000000, 20000002, 200000000, 200000002, 2000000000, 2000000002, 10000000004, 10000100001, 20000000000, 20000000002, 20000000003, 30000000002, 40000000001, but it is not yet ascertained that they are consecutive.

For all terms given above each palindrome is reached from the preceding one or from the start in at most 35 steps; after the presumably last one no further palindrome is reached in 5000 steps.

Additional terms (cf. A090075) are 100000000, 100000001, 100010001, 1000000000, 1000000001, 10000000000, 10000000001, 100000000000, 100000000001, 1000000000000, 1000000000001, 1000001000001, 1000100010001, but it is not yet ascertained that they are consecutive.

For all terms given above each palindrome is reached from the preceding one or from the start in at most 35 steps; after the presumably last one no further palindrome is reached in 5000 steps.

Only two numbers are known whose Reverse and Add trajectory contains twelve palindromes: 10000 and 10001. It is conjectured that these are the only such numbers and it has been conjectured before (cf. A077594) that no Reverse and Add trajectory contains more than twelve palindromes.