[ *February 26, 2001* ]

Palindromes in factorial base and base 10

Find palindromes that remain palindromic when written in factorial base.

Erich Friedman worked on a palindromic subset and submitted the first few terms

to Sloane's database as sequence A046807.

I think it is not a difficult programming exercice to find more terms.

My special interest goes to those numbers which are at the same time

palindromic in __factorial base__ and in __base 10__.

Can you extend the list beyond palindrome 121 ?

**1** = **1** . **1!** = **1**

**3** = **1** . **2!** + **1** . **1!** = **11**

**7** = **1** . **3!** + **0** . **2!** + **1** . **1!** = **101**

**9** = **1** . **3!** + **1** . **2!** + **1** . **1!** = **111**

**11** = **1** . **3!** + **2** . **2!** + **1** . **1!** = **121**

**33** = **1** . **4!** + **1** . **3!** + **1** . **2!** + **1** . **1!** = **1111**

**121** = **1** . **5!** + **0** . **4!** + **0** . **3!** + **0** . **2!** + **1** . **1!** = **10001**

P.S. *Every integer has a unique representation in factorial base*.

This is not apparent as for instance the integer 49 can be written like

**49** = **1** . **4!** + **3** . **3!** + **3** . **2!** + **1** . **1!** = **1331**

or like

**49** = **2** . **4!** + **0** . **3!** + **0** . **2!** + **1**
. **1!** = **2001**

The first (palindromic) expression is invalid though, as the second digit **3** is

greater than the factorial base value **2**!. This is not allowed ! In general :

For every positive integer k there exists a unique sequence of nonnegative integers

d_{1}, d_{2}, ..., d_{n}, (where d_{j} ≤ j for all j) such that

k = d_{1} ·1! + d_{2} ·2! + d_{3} ·3! + ... + d_{n} ·n!

Related websources :

The Factorial Number System.

A007623 - Integers written in factorial base.

Base factorial y otras parecidas

Problem 4: Number Representation in the Factorial Base