The multiplicative persistence of a nonnegative integer 
is the number of steps needed to reach a single digit by
repeatedly replacing the number by the product of its
decimal digits. The final digit
is its multiplicative digital root.
A single digit has multiplicative persistence 0.
For example, the sequence obtained from the starting number 9876 is (9876, 3024, 0), so 9876 has a 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 
.
Brier et al. (2021) proved that, in base 10, a number whose multiplicative digital root is 1, 3, 7, or 9 has multiplicative persistence at most 1, while a number whose multiplicative digital root is 5 has multiplicative persistence at most 5.
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,
with the first term changed from 1 to 0 to count no steps for a single digit;
Gottlieb 1969, 1970, Beeler 1972). 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 multisets of 
digits, excluding the all-zero multiset, is 
. For 
, 2, 3, ..., this gives 9, 54, 219, 714, 2001, 5004, 11439,
... (OEIS A035927). Such a multiset
represents a number up to rearrangement of its digits.
The concept of multiplicative persistence can be generalized to multiplying the 
th
powers of the digits of a number
for an integer 
and iterating until the result is 0 or 1. All numbers
other than repunits, which converge to 1, converge to
0. The number of these iterations needed to reach 0 or 1 is called the 
-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 (Guy 1994).
The smallest primes with multiplicative persistences 
,
1, 2, ... are 2, 11, 29, 47, 277, 769, 8867, 186889, 2678789, 26899889, 3778888999,
277777788888989, ... (OEIS A046500).
