Strictly non-palindromic number

A strictly non-palindromic number is a natural number that is not a number palindrome in any place value system , the basis of which is in the range . ${\displaystyle n}$ ${\displaystyle b}$ ${\displaystyle 2\leq b\leq n-2}$

The upper limit on the size of the base is necessary to keep the sequence nontrivial, since ${\displaystyle n-2}$

• each number (greater than 1) is written as a single-digit (also palindromic) number for each base ; ${\displaystyle n}$ ${\displaystyle b>n}$
• every number (greater than 2) to the base is written as , i.e. non-palindromic; ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle 10}$
• every number (greater than 3) to the base is written as (palindromic). ${\displaystyle n}$ ${\displaystyle n-1}$ ${\displaystyle 11}$

For the set of bases is empty, so these numbers are trivially also strictly non-palindromic. ${\displaystyle n\leq 3}$

Examples

For example, the (decimal) number 6 is written

• to base two: 110,
• to base three: 20 and
• to base four: 12

Since none of these spellings are palindromic, 6 is strictly non-palindromic.

The sequence of strictly non-palindromic numbers begins with

0, 1, 2, 3, 4, 6, 11, 19, 47, 53, 79, 103, 137, 139, 149, 163, 167, 179, 223, 263, 269, 283, 293, ...

properties

All strictly non-palindromic numbers greater than 6 are prime numbers . For every composite number , a base can be found that is palindromic. ${\displaystyle n>6}$ ${\displaystyle n}$

proof

1. If is even , then the base is written as 22 (palindromic). ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle {\tfrac {n}{2}}-1}$
2. Otherwise is odd and can be written as , where the smallest prime factor is. Understandably then . ${\displaystyle n}$ ${\displaystyle n=p\cdot m}$ ${\displaystyle p}$ ${\displaystyle n}$ ${\displaystyle p\leq m}$

• If then , then is what is written for base 2 as 1001 (palindromic). ${\displaystyle p=m=3}$ ${\displaystyle n=9}$
• If then , the base is written as 121 (palindromic). ${\displaystyle p=m>3}$ ${\displaystyle n}$ ${\displaystyle p-1}$

Otherwise is . The case cannot occur because both and are odd. ${\displaystyle p<m-1}$ ${\displaystyle p=m-1}$ ${\displaystyle p}$ ${\displaystyle m}$

In this case, the two-digit number (palindromic) is written to the base . ${\displaystyle n}$ ${\displaystyle {\rm {pp}}}$ ${\displaystyle m-1}$

In each of these cases the base is in the range . ${\displaystyle b}$ ${\displaystyle 2\leq b\leq n-2}$

Individual evidence

1. ↑ Follow A016038 in OEIS