A strictly non-palindromic number is an integer n that is not palindromic in any numeral system with a base b in the range 2 ≤ b ≤ n − 2. For example, the number six is written as 110 in base 2, 20 in base 3 and 12 in base 4, none of which is a palindrome—so 6 is strictly non-palindromic.
The sequence of strictly non-palindromic numbers (sequence A016038 in OEIS) starts:
- 1, 2, 3, 4, 6, 11, 19, 47, 53, 79, 103, 137, 139, 149, 163, 167, 179, 223, 263, 269, 283, 293, …
To test whether a number n is strictly non-palindromic, it must be verified that n is non-palindromic in all bases up to n − 2. The reasons for this upper limit are:
- any n ≥ 3 is written 11 in base n − 1, so n is palindromic in base n − 1;
- any n ≥ 2 is written 10 in base n, so any n is non-palindromic in base n;
- any n ≥ 1 is a single-digit number in any base b > n, so any n 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.
Read more about Strictly Non-palindromic Number: Properties
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