Multiplicative Persistence and Multiplicative Digital Root

My diurnal age today (26748) has the property that it has a multiplicative persistence of 6. This qualifies it for membership in OEIS A199996:

A199996

Composite numbers whose multiplicative persistence is 6.

The initial members of the sequence are:

6788, 6878, 6887, 7688, 7868, 7886, 8678, 8687, 8768, 8786, 8876, 16788, 16878, 16887, 17688, 17868, 17886, 18678, 18687, 18768, 18786, 18867, 18876, 23788, 23878, 24678, 24687, 24768, 24786, 24867, 24876, 26478, 26487, 26748, 26784, 26847, 26874, 27388 ...

To quote from Wikipedia:

In number theory, the multiplicative digital root of a natural number \(n\) in a given number base \(b\) is found by multiplying the digits of \(n\) together, then repeating this operation until only a single-digit remains, which is called the multiplicative digital root of \(n\). Multiplicative digital roots are the multiplicative equivalent of digital roots.

The number of iterations required to reach the multiplicative digital root is termed the multiplicative persistence. It is conjectured that there is no number with a multiplicative persistence greater than 11. The smallest numbers with multiplicative persistence of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 are:

0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899

These numbers constitute OEIS A003001 in the OEIS (see permalink for an algorithm that will generate the members of this sequence up to one million). 

Here is permalink to an algorithm that will calculate multiplicative persistence and multiplicative digital roots for a range of numbers (both composite and prime). The algorithm is easily modified to search for a specific multiplicative persistence or multiplicative digital root. Primes can be excluded by simply adding that condition to the relevant section of the code.

ADDENDUM

On October 4th 2022, my diurnal age was 26847, a permutation of the digits of 26748 and thus also having a multiplicative persistence of 6. In between these two diurnal ages, I passed 26784 days, another permutation, and 26874 is still to come. Altogether there are 120 permutations of the digits 2, 4, 6, 7 and 8. 

It's interesting to look at a breakdown of the percentages of numbers with various multiplicative persistences. Up to one million, the breakdown is:

7    0.245%

6    0.449%

5    2.47%

4    6.68%

3    12.4%

2    37.5%

1    40.3%

In this range there are no numbers with a multiplicative persistence of 8. The first such number is 2,677,889.