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:
“For love ... has two faces; one white, the other black; two bodies; one smooth, the other hairy. It has two hands, two feet, two tails, two, indeed, of every member and each one is the exact opposite of the other. Yet, so strictly are they joined together that you cannot separate them.”
—Virginia Woolf (1882–1941)
“No Government can be long secure without a formidable Opposition. It reduces their supporters to that tractable number which can be managed by the joint influences of fruition and hope. It offers vengeance to the discontented, and distinction to the ambitious; and employs the energies of aspiring spirits, who otherwise may prove traitors in a division or assassins in a debate.”
—Benjamin Disraeli (1804–1881)