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.
A representation of a number n in base b, where b > 1 and n > 0, is a sequence of k+1 digits ai (0 ≤ i ≤ k) such that
and 0 ≤ ai < b for all i and ak ≠ 0.
Such a representation is defined as palindromic if ai = ak−i 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 (sequence A016038 in the OEIS) starts:
For example, the number 19 written in base 2 through 17 is:
|19 in base b||10011||201||103||34||31||25||23||21||19||18||17||16||15||14||13||12|
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.
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.