“Multiplicative Persistence”的概念、定义、习题解题方法及翻译-数学辞典

Multiply all the digits of a number

by each other, repeating with the product until a single Digit is obtained. Thenumber of steps required is known as the multiplicative persistence, and the final Digit obtained is called theMultiplicative Digital Root of

For example, the sequence obtained from the starting number 9876 is (9876, 3024, 0), so 9876 has an multiplicativepersistence of two and a Multiplicative Digital Root of 0. The multiplicative persistences of the first few positiveintegers 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 withpersistence 11 is

The maximum multiplicative persistence in base 2 is 1. It is conjectured that all powers of 2

contain a 0 inbase 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. Themaximum 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 anditerating until the result remains constant. All numbers other than Repunits, which converge to 1, convergeto 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 positiveintegers.

	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, ...

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, andA003001/M4687in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.