Multiplicative Persistence

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, ... (OEIS A031346). The smallest numbers having multiplicative persistences of 1, 2, ... are 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899, ... (OEIS A003001; Wells 1986, p. 78). There is no number  with multiplicative persistence  (Carmody 2001; updating Wells 1986, p. 78). 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 number length. The maximum lengths for -, 2-, 3-, ..., digit numbers are 0, 4, 5, 6, 7, 7, 8, 9, 9, 10, 10, 10, ... (OEIS A014553; Beeler 1972; Gottlieb 1969, 1970). The numbers of -digit numbers having maximal multiplicative persistence for , 2, ..., are 10 (which includes the number 0), 1, 9, 12, 20, 2430, ... (OEIS A046148). The smallest -digit numbers with maximal multiplicative persistence are 0, 77, 679, 6788, 68889, 168889, ... (OEIS A046149). The largest -digit numbers with maximal multiplicative persistence are 9, 77, 976, 8876, 98886, 997762, ... (OEIS A046150). The number of distinct -digit numbers (except for 0s) are given by  which, for , 2, 3, ..., gives 54, 219, 714, 2001, 5004, 11439, ... (OEIS A035927).

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	A031348	0, 7, 6, 6, 3, 5, 5, 4, 5, 1, ...
3	A031349	0, 4, 5, 4, 3, 4, 4, 3, 3, 1, ...
4	A031350	0, 4, 3, 3, 3, 3, 2, 2, 3, 1, ...
5	A031351	0, 4, 4, 2, 3, 3, 2, 3, 2, 1, ...
6	A031352	0, 3, 3, 2, 3, 3, 3, 3, 3, 1, ...
7	A031353	0, 4, 3, 3, 3, 3, 3, 2, 3, 1, ...
8	A031354	0, 3, 3, 3, 2, 4, 2, 3, 2, 1, ...
9	A031355	0, 3, 3, 3, 3, 2, 2, 3, 2, 1, ...
10	A031356	0, 2, 2, 2, 3, 2, 3, 2, 2, 1, ...

Erdős suggested ignoring all zeros and showed that at most  steps are needed to reduce  to a single digit, where  depends on the base.

The smallest primes with multiplicative persistences , 2, 3, ... are 2, 29, 47, 277, 769, 8867, 186889, 2678789, 26899889, 3778888999, 277777788888989, ... (OEIS A046500).

REFERENCES:

Beeler, M. Item 56 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 22, Feb. 1972. https://www.inwap.com/pdp10/hbaker/hakmem/number.html#item56.

Carmody, P. "OEIS A003001, and a 'Zero-Length Message'." 23 Jul 2001. https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0107&L=NMBRTHRY&P=R1036&I=-3.

Gardner, M. Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 170 and 186, 1992.

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.

Pickover, C. A. "Persistence." Ch. 28 in Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning. Oxford, England: Oxford University Press, 2001.

Rivera, C. "Problems & Puzzles: Puzzle 022-Primes & Persistence." https://www.primepuzzles.net/puzzles/puzz_022.htm.

Schneider, W. "The Persistence of a Number." https://www.wschnei.de/digit-related-numbers/persistence.html.

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

Sloane, N. J. A. Sequences A003001/M4687, A014553, A031346, and A046500 in "The On-Line Encyclopedia of Integer Sequences."

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 78, 1986.