The terms SOD and POD are used here to refer to Sum Of Digits and Product Of Digits. I've made a post titled SOD ET AL on June 29th 2021. Quite recently on March 10th 2024, I made another post titled Permutations Involving Sum and Product of Digits and like that post, this post involves a combination of SOD and POD.
On March 2nd 2024, I turned 27362 days old and the number 27362 has the following property as noted in my Airtable record:
27362 is a number \(n\) without the digit 0 with two distinct prime factors such that \(n\) + SOD(\(n\)) and \(n\) + POD(\(n\)) both have two distinct prime factors. Here SOD stands for sum of digits and POD for product of digits. Note that this is different to the arithmetic and multiplicative digital roots of a number. Here the results for \(n\), \(n\) + SOD(\(n\)) and \(n\) + POD(\(n\)) are: $$27362 = 2 \times 13681\\27382 = 2 \times 13691\\27866 = 2 \times 13933$$The members of this sequence from 27362 up to 40000 are (permalink):
27362, 27373, 27389, 27395, 27419, 27443, 27493, 27515, 27535, 27571, 27578, 27598, 27635, 27641, 27649, 27757, 27842, 27849, 27899, 27933, 27934, 28141, 28187, 28235, 28293, 28321, 28345, 28369, 28498, 28529, 28769, 28783, 28811, 28846, 28874, 28963, 29219, 29227, 29263, 29278, 29291, 29335, 29377, 29485, 29487, 29534, 29543, 29553, 29593, 29594, 29617, 29626, 29657, 29765, 29773, 29797, 29951, 31187, 31273, 31435, 31439, 31462, 31618, 31619, 31631, 31677, 31693, 31754, 31762, 31767, 31783, 31826, 31874, 31893, 32161, 32177, 32179, 32221, 32449, 32521, 32527, 32534, 32551, 32629, 32666, 32735, 32755, 32819, 32827, 32845, 32863, 32881, 33121, 33133, 33431, 33458, 33499, 33523, 33526, 33643, 33658, 33659, 33671, 33729, 33837, 33842, 33877, 33926, 33947, 33963, 33983, 34315, 34321, 34363, 34467, 34514, 34531, 34555, 34634, 34733, 34754, 34829, 34837, 34873, 34966, 34973, 34993, 35138, 35218, 35219, 35233, 35318, 35366, 35414, 35477, 35522, 35611, 35614, 35633, 35657, 35678, 35693, 35726, 35761, 35782, 35789, 35813, 35857, 35887, 35927, 36111, 36154, 36169, 36178, 36193, 36227, 36289, 36398, 36447, 36463, 36485, 36535, 36577, 36641, 36733, 36759, 36853, 36893, 36961, 37165, 37239, 37381, 37486, 37586, 37615, 37678, 37787, 37837, 37865, 37943, 37981, 38137, 38179, 38243, 38245, 38297, 38359, 38422, 38429, 38463, 38473, 38489, 38515, 38549, 38615, 38758, 38771, 38837, 38849, 38854, 38914, 38926, 38957, 38978, 38983, 38999, 39127, 39145, 39257, 39413, 39453, 39481, 39637, 39661, 39723, 39747, 39811, 39871, 39917, 39941, 39959
We can extend this idea to sphenic numbers and consider numbers \(n\) without the digit 0 with three distinct prime factors such that \(n\) + SOD(\(n\)) and \(n\) + POD(\(n\)) both have three distinct prime factors. An example of such a number is 27544 where \(n\), \(n\) + SOD and \(n\) + POD factorise respectively as follows:$$ \begin{align} 27554 &= 2 \times 23 \times 599\\27577 &= 11 \times 23 \times 109\\28954 &= 2 \times 31 \times 467 \end{align}$$The numbers satisfying this condition from 27544 up to 40000 are:
27554, 27671, 27745, 27813, 27914, 27982, 28118, 28217, 28226, 28326, 28353, 28355, 28366, 28514, 28535, 28713, 28819, 28878, 28954, 29559, 29589, 29622, 29829, 29926, 29955, 29958, 31215, 31274, 31538, 31611, 31623, 31642, 31659, 31726, 31742, 31983, 32151, 32195, 32218, 32241, 32326, 32394, 32421, 32457, 32542, 32631, 32739, 32829, 32862, 32883, 32997, 33226, 33297, 33319, 33341, 33438, 33454, 33586, 33734, 33765, 33882, 33971, 34131, 34143, 34359, 34498, 34539, 34622, 34655, 34683, 34773, 34941, 34953, 34959, 34977, 35165, 35185, 35265, 35371, 35529, 35686, 35866, 35949, 36177, 36249, 36381, 36417, 36534, 36597, 36669, 36698, 36718, 36743, 36933, 37118, 37222, 37247, 37262, 37378, 37383, 37497, 37522, 37542, 38234, 38253, 38337, 38355, 38361, 38395, 38566, 38674, 39219, 39238, 39242, 39263, 39277, 39282, 39369, 39462, 39515, 39538, 39542, 39621, 39639
These sequences do not appear in the OEIS and I certainly won't be submitting them (pearls before swine) but they are interesting examples of sequences arising from a combination of SOD and POD.
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