The multiplicative digital root of a nonnegative integer 
is the single digit obtained by repeatedly replacing the
number by the product of its decimal digits.
The number of replacements required is its multiplicative
persistence. A single digit is already its own multiplicative
digital root and requires no replacements.
For example, the sequence obtained from the starting number 9876 is (9876, 3024, 0), so 9876 has a multiplicative persistence of two and a multiplicative digital root of 0. The multiplicative digital roots of the first few positive integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, ... (OEIS A031347).
 | OEIS | numbers having
multiplicative digital root  |
| 0 | A034048 | 0, 10, 20, 25, 30, 40, 45, 50, 52, 54, 55, 56, 58, ... |
| 1 | A002275 | 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, ... |
| 2 | A034049 | 2, 12, 21, 26, 34, 37, 43, 62, 73, 112, 121, 126, ... |
| 3 | A034050 | 3, 13, 31, 113, 131, 311, 1113, 1131, 1311, 3111, ... |
| 4 | A034051 | 4, 14, 22, 27, 39, 41, 72, 89, 93, 98, 114, 122, ... |
| 5 | A034052 | 5, 15, 35, 51, 53, 57, 75, 115, 135, 151, 153, 157, ... |
| 6 | A034053 | 6, 16, 23, 28, 32, 44, 47, 48, 61, 68, 74, 82, 84, ... |
| 7 | A034054 | 7, 17, 71, 117, 171, 711, 1117, 1171, 1711, 7111, ... |
| 8 | A034055 | 8, 18, 24, 29, 36, 38, 42, 46, 49, 63, 64, 66, 67, ... |
| 9 | A034056 | 9, 19, 33, 91, 119, 133, 191, 313, 331, 911, 1119, ... |
