Strictly Non-palindromic Number

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 ≤ bn − 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:

    Yesterday the Electoral Commission decided not to go behind the papers filed with the Vice-President in the case of Florida.... I read the arguments in the Congressional Record and can’t see how lawyers can differ on the question. But the decision is by a strictly party vote—eight Republicans against seven Democrats! It shows the strength of party ties.
    Rutherford Birchard Hayes (1822–1893)

    In the U.S. for instance, the value of a homemaker’s productive work has been imputed mostly when she was maimed or killed and insurance companies and/or the courts had to calculate the amount to pay her family in damages. Even at that, the rates were mostly pink collar and the big number was attributed to the husband’s pain and suffering.
    Gloria Steinem (20th century)