[ 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
d₁, d₂, ..., d_n, (where d_j ≤ j for all j) such that
k = d₁ ·1! + d₂ ·2! + d₃ ·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