Strictly non-palindromic number
A strictly non-palindromic number is an integer "n" that is not palindromic in any positional numeral system with a base "b" in the range 2 ≤ "b" ≤ "n" − 2. For example, the number 6 is written as "110" in base 2, "20" in base 3 and "12" in base 4, none of which are palindromes—so 6 is strictly non-palindromic. Definition. A representation of a number "n" in base "b", where "b" > 1 and "n" > 0, is a sequence of "k"+1 digits "a" (0 ≤ "i" ≤ "k") such that  and 0 ≤ "a" < "b" for all "i" and "a" ≠ 0. Such a representation is defined as palindromic if "a" = "a" for all "i". A number "n" is defined as strictly non-palindromic if the representation of "n" is not palindromic any base "b" where 2 ≤ "b" ≤ "n"-2. The sequence of strictly non-palindromic numbers starts: For example, the number 19 written in base 2 through 17 is: None of these is a palindrome, so 19 is a strictly non-palindromic number. The reason for the upper limit of "n" − 2 on the base  is that all numbers are trivially palindromic in large bases: Thus it can be seen that the upper limit of "n" − 2 is necessary to obtain a mathematically "interesting" definition. For "n" < 4 the range of bases is empty, so these numbers are strictly non-palindromic in a trivial way. Properties. All strictly non-palindromic numbers larger than 6 are prime. One can prove that a composite "n" > 6 cannot be strictly non-palindromic as follows. For each such "n" a base is shown to exist in which "n" is palindromic. Strictly non-palindromic number A