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).

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