Multiplicative and Additive Digital Roots

Even though I've written about multiplicative and 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.