Intrinsic k-palindromes

[ *May 20, 2001* ]

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 EIS 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 **111**_{3} and is of length **3**

**666** is **5**-palindromic because it equals to **22122**_{4} and is of length **5**

**1667** is **7**-palindromic because it equals to **2021202**_{3} and is of length **7**

**511** is **9**-palindromic because it equals to **111111111**_{2} and is of length **9**

**180388** is **12**-palindromic because it equals to **100011110001**_{3} and is of length **12**