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
Famous quotes containing the words strictly and/or number:
“You are not satisfied unless form is so strictly divorced from content that you can comprehend the one without almost without bothering to read the other.”
—Samuel Beckett (1906–1989)
“My idea is that the world outside—the so-called modern world—can only pervert and degrade the conceptions of the primitive instinct of art and feeling, and that our only chance is to accept the limited number of survivors—the one- in-a-thousand of born artists and poets—and to intensify the energy of feeling within that radiant centre.”
—Henry Brooks Adams (1838–1918)