Consider the process of taking a number, multiplying its digits, then multiplying the digits of numbers derived from it, etc., until the remaining number has only one digit. The number of multiplications required to obtain a single digit from a number is called the multiplicative persistence of , and the digit obtained is called the multiplicative digital root of .

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, ... |