A286138 Pseudo-palindromic numbers: not palindromes (A002113), but a nontrivial palindromic concatenation (AA or ABA) of arbitrary nonzero integers A and B.

1010, 1101, 1121, 1131, 1141, 1151, 1161, 1171, 1181, 1191, 1201, 1211, 1212, 1231, 1241, 1251, 1261, 1271, 1281, 1291, 1301, 1311, 1313, 1321, 1341, 1351, 1361, 1371, 1381, 1391, 1401, 1411, 1414, 1421, 1431, 1451, 1461, 1471, 1481, 1491, 1501, 1511, 1515, 1521, 1531

Offset: 1

Author

M. F. Hasler, May 03 2017

Keywords

• nonn
• base

Comments

The pseudo- or almost-palindromic numbers considered here are not related to the similarly named but different concepts mentioned in comments on A003555 and in A060087 - A060088.

We could consider "more general" palindromic concatenations like A.B.B.A, A.B.C.B.A, etc., but all of these can be written as A.B'.A with B' = B.B resp. B.C.B, etc. The result is non-palindromic (i.e., not in A002113) as required, if and only if at least one of the strings is non-palindromic.

Here, A is allowed to have only one digit, so most of the first 100 terms are of the form 1.B.1 where B = 10, 12, 13, ... (palindromes 11, 22, 33, ... excluded).

If all of the strings A, B (...) are required to be non-palindromic, the sequence starts with terms of the form A.A with A = 10, 12, 13, ..., 98: 1010, 1212, 1313, 1414, 1515, 1616, 1717, 1818, 1919, 2020, 2121, 2323, .... This is a subsequence of A239019 (numbers which are not primitive words over the alphabet {0,...,9} when written in base 10).

Programs

PARI

A286138 = select(t->!is_A002113(t),setunion(vector(801,i,((i-1)\89+1)*1001+((i-1)%89+1)*10),vector(89,i,(i+9)*101))) \\ The first 810 terms.