WON plate92 | World!OfNumbers Palindromes in factorial base and base 10. [ February 26, 2001 ] 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, ..., dn, (where dj <= j for all j) such that k = d1 ·1! + d2 ·2! + d3 ·3! + ... + dn ·n! A000092 Prime Curios! Prime Puzzle Wikipedia 92 Le Nombre 92 Numberland 92
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