Multiplicative and Additive Digital Roots

Even though I've written about multiplicative and arithmetic digital roots in numerous posts, it would seem that I've never addressed the obvious question of how many numbers have identical roots. I was searching for properties of the number associated with my diurnal age (28131) when I noticed the following:$$ \begin{align} 28131 &\rightarrow 2 + 8 + 1+3+1 = 15 \rightarrow 1 + 5 =6 \\ 28131 &\rightarrow 2 \times 8 \times 1 \times 3 \times 1 =48 \rightarrow 4 \times 8 =32 \rightarrow 3 \times 2 = 6 \end{align}$$It turns out that there are \( \textbf{1085} \) such numbers in the range between 1 and 40000, representing 2.7125% of the range. I won't list all of the numbers here but only those from my diurnal age up to 40000 (permalink):

28131, 28167, 28169, 28176, 28178, 28187, 28196, 28223, 28232, 28311, 28322, 28347, 28374, 28437, 28473, 28617, 28619, 28671, 28691, 28716, 28718, 28734, 28743, 28761, 28781, 28817, 28871, 28916, 28961, 29117, 29126, 29162, 29168, 29171, 29186, 29216, 29261, 29612, 29618, 29621, 29681, 29711, 29816, 29861, 29999, 31113, 31128, 31131, 31139, 31169, 31182, 31193, 31196, 31218, 31227, 31234, 31243, 31272, 31281, 31311, 31319, 31324, 31342, 31344, 31391, 31423, 31432, 31434, 31443, 31619, 31677, 31691, 31722, 31767, 31776, 31778, 31787, 31812, 31821, 31877, 31889, 31898, 31913, 31916, 31931, 31961, 31988, 32118, 32127, 32134, 32143, 32172, 32181, 32217, 32226, 32228, 32262, 32271, 32282, 32314, 32336, 32341, 32363, 32413, 32431, 32478, 32487, 32622, 32633, 32712, 32721, 32748, 32784, 32811, 32822, 32847, 32874, 33111, 33119, 33124, 33142, 33144, 33191, 33214, 33236, 33241, 33263, 33326, 33344, 33362, 33412, 33414, 33421, 33434, 33441, 33443, 33477, 33479, 33497, 33557, 33575, 33623, 33632, 33666, 33747, 33749, 33755, 33774, 33794, 33911, 33947, 33974, 34123, 34132, 34134, 34143, 34213, 34231, 34278, 34287, 34312, 34314, 34321, 34334, 34341, 34343, 34377, 34379, 34397, 34413, 34431, 34433, 34728, 34737, 34739, 34773, 34782, 34793, 34827, 34872, 34937, 34973, 35357, 35375, 35537, 35573, 35735, 35753, 36119, 36177, 36191, 36222, 36233, 36323, 36332, 36366, 36636, 36663, 36717, 36771, 36911, 37122, 37167, 37176, 37178, 37187, 37212, 37221, 37248, 37284, 37347, 37349, 37355, 37374, 37394, 37428, 37437, 37439, 37473, 37482, 37493, 37535, 37553, 37617, 37671, 37716, 37718, 37734, 37743, 37761, 37781, 37817, 37824, 37842, 37871, 37934, 37943, 38112, 38121, 38177, 38189, 38198, 38211, 38222, 38247, 38274, 38427, 38472, 38717, 38724, 38742, 38771, 38819, 38891, 38918, 38981, 39113, 39116, 39131, 39161, 39188, 39311, 39347, 39374, 39437, 39473, 39611, 39734, 39743, 39818, 39881

All permutations of any of these numbers will have multiplicative and arithmetic digital roots that are the same. Putting the digits of 28131 in ascending order, we get 11238. If we only consider numbers whose digits are in ascending order, then in the range up to 40000 there are only \( \textbf{74}\) numbers that qualify. These are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 123, 137, 139, 168, 179, 188, 233, 267, 299, 346, 389, 899, 1124, 1157, 1347, 1355, 1469, 1779, 1788, 2236, 2346, 2348, 2778, 3335, 3779, 11126, 11133, 11148, 11177, 11222, 11238, 11279, 11339, 11369, 11579, 11666, 11677, 11679, 11699, 11999, 12237, 12269, 12334, 12444, 12446, 12678, 12689, 12777, 12788, 13344, 13677, 13778, 13889, 14777, 22236, 22238, 23336, 23478, 29999, 33344, 33477, 33479, 33557, 33666

Permutations of the digits of these numbers will generate the other 1011 (1085 - 74) numbers in the range. These numbers are members of OEIS A064702.

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.