WON plate101 | World!OfNumbers [ May 20, 2001 ] Intrinsic k-palindromes We define a number N k-palindromic if there is a base b such that N is a palindrome of length k in base b. Neil Sloane found this notion of palindromicity of a natural number which is independent of the base described in the following reference : A. J. Di Scala and M. Sombra, Intrinsic palindromic numbers They show that all numbers are 1- and 2-palindromes, most are 3-, while most are not 4- and higher k-palindromes. Harvey P. Dale kindly worked out the sequences of the first values of k for the OEIS database : Intrinsic 3-palindromes A060873 Intrinsic 4-palindromes A060874 Intrinsic 5-palindromes A060875 Intrinsic 6-palindromes A060876 Intrinsic 7-palindromes A060877 Intrinsic 8-palindromes A060878 Intrinsic 9-palindromes A060879 Intrinsic 10-palindromes A060947 Intrinsic 11-palindromes A060948 Intrinsic 12-palindromes A060949 A few examples 13 is 3-palindromic because it equals to 1113 and is of length 3 666 is 5-palindromic because it equals to 221224 and is of length 5 1667 is 7-palindromic because it equals to 20212023 and is of length 7 511 is 9-palindromic because it equals to 1111111112 and is of length 9 180388 is 12-palindromic because it equals to 1000111100013 and is of length 12 A000101 Prime Curios! Prime Puzzle Wikipedia 101 Le nombre 101
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