Some Special Sphenic Numbers

Today I turned $\mathbf{\text{27782}}$ days old and one of the properties of 27782 is that it is sphenic since:$$27782=2\times 29\times 479$$However, looking at the digital roots of the number and its factor we notice an interesting fact:$$\underset{8}{\underbrace{27782}}=\underset{2}{\underbrace{2}}\times \underset{2}{\underbrace{29}}\times \underset{2}{\underbrace{479}}$$The respective digital roots are shown under the number and its factors and we see that the digital roots of the factors (2) multiply together to give the digital root of the number (8). The only other way this can occur is if all the digital roots are 1.

In the range up to 100,000 there are 230 sphenic numbers with the property that the digital roots of the factors multiply to give the digital root of the number. These numbers are (permalink):

638, 1034, 1826, 2222, 2726, 3014, 3806, 4202, 4814, 4994, 5786, 5858, 6182, 6974, 7766, 7802, 7946, 8558, 9494, 9746, 10034, 10142, 10538, 11078, 12518, 12878, 12914, 13166, 14102, 14498, 14894, 14993, 15254, 16262, 16298, 16766, 17954, 18062, 18386, 18458, 18854, 20042, 20438, 20474, 20834, 21338, 21626, 22418, 22562, 22742, 24002, 24398, 24722, 25586, 25694, 25982, 26414, 26477, 26738, 26774, 27674, 27782, 28358, 28718, 28754, 29798, 29942, 31526, 31706, 31922, 32219, 32714, 33002, 33182, 33506, 34046, 34298, 34946, 35486, 36566, 36674, 37178, 37682, 37862, 38222, 38582, 39266, 39842, 40634, 41642, 41822, 42911, 43334, 43406, 43658, 43703, 44594, 45026, 45386, 45782, 45854, 46178, 46646, 47366, 47402, 47618, 48554, 48662, 49346, 49706, 50534, 51319, 51326, 51722, 52217, 52334, 52622, 52838, 53126, 53306, 53486, 53702, 53882, 54098, 54494, 54926, 55178, 55187, 55682, 56078, 56762, 57014, 57662, 58454, 58598, 59102, 59246, 59642, 60038, 60254, 60929, 61622, 61946, 62018, 62198, 62414, 63278, 63638, 63998, 64034, 64322, 64394, 64574, 65186, 65978, 66086, 67454, 67958, 68498, 70586, 70829, 71306, 71522, 72062, 72413, 73106, 73538, 73898, 74762, 75086, 75806, 75878, 76274, 76526, 76627, 76994, 77174, 77858, 78254, 78542, 78578, 78866, 78938, 79046, 79514, 80558, 81422, 81818, 83114, 83897, 84158, 84986, 85382, 85634, 86174, 86246, 86714, 86858, 87326, 87758, 88154, 88334, 89018, 89281, 89441, 89486, 89639, 89738, 90422, 90926, 90998, 92213, 92402, 93122, 93302, 93554, 93698, 94454, 95678, 95786, 96686, 96722, 96758, 97226, 97262, 97442, 98054, 98747, 98846, 99818

However, of these only three have digital roots that are all equal to 1. These are:$$\begin{array}{rl}51319& =19\times 37\times 73\\ 76627& =19\times 37\times 109\\ 89281& =19\times 37\times 127\end{array}$$As can be seen, two of the factors (19 and 37) are the same for all three numbers. If we extend the range to one million, these two factor and 73 (the reversal of 37) make frequent appearances.

If we relax the requirement that the digital roots of the factors must be equal and require only the the digital roots of the factors multiply together to give the digital root of the number, then we find that 1969 numbers satisfy and that will include the 230 numbers mentioned earlier (permalink). An example would be 27813:$$\underset{3}{\underbrace{27813}}=\underset{3}{\underbrace{3}}\times \underset{1}{\underbrace{73}}\times \underset{1}{\underbrace{127}}$$