Multiply all the digits of a number by each other, repeating with the product until a single Digit is obtained. The number of steps required is known as the multiplicative persistence, and the final 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 an multiplicative
persistence of two and a Multiplicative Digital Root of 0. The multiplicative persistences of the first few positive
integers are 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2,
2, 3, 1, 1, ... (Sloane's A031346). The smallest numbers having multiplicative persistences of 1, 2, ... are 10, 25,
39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899, ... (Sloane's A003001). There is no number
with multiplicative persistence . It is conjectured that the maximum number lacking the Digit 1 with
persistence 11 is

There is a stronger conjecture that there is a maximum number lacking the Digit 1 for each persistence .

The maximum multiplicative persistence in base 2 is 1. It is conjectured that all powers of 2 contain a 0 in base 3, which would imply that the maximum persistence in base 3 is 3 (Guy 1994).

The multiplicative persistence of an -Digit number is also called its Length. The
maximum lengths for -, 3-, ..., digit numbers are 4, 5, 6, 7, 7, 8, 9, 9, 10, 10, 10, ...
(Sloane's A014553; (Beeler *et al. *1972, Item 56; Gottlieb 1969-1970).

The concept of multiplicative persistence can be generalized to multiplying the th powers of the digits of a number and iterating until the result remains constant. All numbers other than Repunits, which converge to 1, converge to 0. The number of iterations required for the th powers of a number's digits to converge to 0 is called its -multiplicative persistence. The following table gives the -multiplicative persistences for the first few positive integers.

Sloane | -Persistences | |

2 | Sloane's A031348 | 0, 7, 6, 6, 3, 5, 5, 4, 5, 1, ... |

3 | Sloane's A031349 | 0, 4, 5, 4, 3, 4, 4, 3, 3, 1, ... |

4 | Sloane's A031350 | 0, 4, 3, 3, 3, 3, 2, 2, 3, 1, ... |

5 | Sloane's A031351 | 0, 4, 4, 2, 3, 3, 2, 3, 2, 1, ... |

6 | Sloane's A031352 | 0, 3, 3, 2, 3, 3, 3, 3, 3, 1, ... |

7 | Sloane's A031353 | 0, 4, 3, 3, 3, 3, 3, 2, 3, 1, ... |

8 | Sloane's A031354 | 0, 3, 3, 3, 2, 4, 2, 3, 2, 1, ... |

9 | Sloane's A031355 | 0, 3, 3, 3, 3, 2, 2, 3, 2, 1, ... |

10 | Sloane's A031356 | 0, 2, 2, 2, 3, 2, 3, 2, 2, 1, ... |

**References**

Beeler, M.; Gosper, R. W.; and Schroeppel, R. *HAKMEM.* Cambridge, MA: MIT
Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

Gottlieb, A. J. Problems 28-29 in ``Bridge, Group Theory, and a Jigsaw Puzzle.'' *Techn. Rev.* **72**, unpaginated, Dec. 1969.

Gottlieb, A. J. Problem 29 in ``Integral Solutions, Ladders, and Pentagons.'' *Techn. Rev.* **72**, unpaginated, Apr. 1970.

Guy, R. K. ``The Persistence of a Number.'' §F25 in
*Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 262-263, 1994.

Sloane, N. J. A. ``The Persistence of a Number.'' *J. Recr. Math.* **6**, 97-98, 1973.

Sloane, N. J. A. A014553, A031346, and A003001/M4687 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

© 1996-9

1999-05-26