A strictly non-palindromic number is an integer n that is not palindromic in any positional numeral system with a baseb 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 baseb, where b > 1 and n > 0, is a sequence of k+1 digitsai (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:
b
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
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:
In base b = n − 1, n ≥ 3 is written "11".
In any base b > n, n is a single digit, so is palindromic in all such 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[]
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All strictly non-palindromic numbers larger than 6 are prime. One can prove that a compositen > 6 cannot be strictly non-palindromic as follows. For each such n a base is shown to exist in which n is palindromic.
If n is even and greater than 6, then n is written "22" (a palindrome) in base n/2 − 1. (Note that if n less than or equal to 6, the base n/2 − 1 would be less than 3, so the digit "2" could not occur in the representation of n.)
If n is odd and greater than 1, write n = p · m, where p is the smallest prime factor of n. Clearly p ≤ m (since n is composite).
If p = m (that is, n = p2), there are two cases:
If p = 3, then n = 9 is written "1001" (a palindrome) in base 2.
If p > 3, then n is written "121" (a palindrome) in base p − 1.
p cannot equal m - 1 because both p and m are odd, so p < m − 1. Then n can be written as the two-digit number pp in base m − 1.
