I have written a small program that calculates the base representation of a number, and checks if that representation is a palindrome (also check if the length is at least 2, since a length of 1 is trivially palindromic).
What I have found is that there is a "gap", which becomes larger as the numbers that are considered goes higher. In the image below, I have considered all numbers from 2 to 30000. There also seems to be 2 smaller gaps forming in between each of the "segments".
Is there a reason that there is a gap?
Edit: the condition on being palindromic are considering the concatenation of the decimal representations using code found here: https://stackoverflow.com/questions/10184577/java-recursion-program-convert-a-base-10-number-to-any-base. For example, I consider 5999 palindromic in base 111, because its representation is 54 : 5 (base 111), and the concatenation of the digits (545) is palindromic.
Here is considering all numbers $\le 3*10^4$:
Here is considering all numbers $\le 10^5$: