[ 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