cp's OEIS Frontend

Only positive palindromes are considered.

The numbers N(n) of distinct types of n-digit palindromes, for n=1,2,..., are 1,1,2,2,5,5,15,15,... (A164904, n>=1). It is easy to see that N(2*n-1)=N(2*n), n>=1.

The sequence is related to A266991.

Sequence {A164864(n) - a(n)}_(n>=1) begins 0,0,1,4,1,26,1,175,365,1516,...

One can explain, why, for example, a(4)=11, instead of A164864(4)=15. There exist exactly 4 types of 4-digit numbers, which cannot be prime. In A266946 these types are: 1001, 1010, 1100, 1111. Indeed, numbers abba,aabb,aaaa are divisible by 11; a number abab is divisible by 101.

In other cases of n-digit types we should verify the divisibility of numbers of types in A266946 at least by primes of the form 11,101,... Besides, a digital type 1...1 exists only for n in A004023, i.e., for only 9 values of n from the first 270343. This simplifies the calculations.

a(n) <= A376918(n) with equality for n <= 9, but thereafter some digital types which pass the divisibility rules of A376918 don't in fact occur among the primes (see A377727). - Dmytro Inosov, Nov 05 2024

Based on the conjectured terms in A377727, the next three terms can be conjectured: a(14)=190402538; a(15)=1378294708; a(16)=10437142874. - Dmytro Inosov, Jan 07 2025

Permuting the symbols does not change the structure; so e.g. aba and bab are equivalent.