Multiplicative Persistence

Multiply all the digits of a number $n$ 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 $n$.

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 $<10^{50}$ with multiplicative persistence $>11$. It is conjectured that the maximum number lacking the Digit 1 with persistence 11 is

$$77777733332222222222222222222$$

There is a stronger conjecture that there is a maximum number lacking the Digit 1 for each persistence $\geq 2$.

The maximum multiplicative persistence in base 2 is 1. It is conjectured that all powers of 2 $>2^{15}$ 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 $n$-Digit number is also called its Length. The maximum lengths for $n=2$-, 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 $k$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 $k$th powers of a number's digits to converge to 0 is called its $k$-multiplicative persistence. The following table gives the $n$-multiplicative persistences for the first few positive integers.

$n$	Sloane	$n$-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, ...

See also 196-Algorithm, Additive Persistence, Digitaddition, Digital Root, Kaprekar Number, Length (Number), Multiplicative Digital Root, Narcissistic Number, Recurring Digital Invariant

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.