Sum of Divisors Meets the Totient

The number associated with my diurnal age today, \(\textbf{27838}\), has an interesting property that is not so obvious until we look at its sum of divisors and it totient:$$ \begin{align}  \sigma(27838) &= 43200 \\ &= 2^6 \times 3^3 \times 5^2 \\ &\rightarrow 2,3,5 \text{ as distinct prime factors}\\ \phi(27838) &= 13440 \\&= 2^7 \times 3 \times 5 \times 7 \\ &\rightarrow 2,3,5,7 \text{ as distinct prime factors} \end{align}$$For both the sum of divisors and the totient, the prime factors are consecutive. This got me thinking as to how many numbers enjoy this property in the range up to 40000. I wasn't requiring that the smallest factor be 2 for both the sum of divisors and the totient but this is certainly the case at least in the range under consideration. 

In developing my algorithm (permalink), I naturally only considered composite numbers but I also required the sum of divisors and the totient to be composite as well. It turns out that there are 785 such numbers with the smallest being 14:$$ \begin{align}  \sigma(14) &= 24 \\ &= 2^3 \times 3 \\ &\rightarrow 2,3 \text{ as distinct prime factors}\\ \phi(14) &= 6 \\&= 2 \times 3  \\ &\rightarrow 2,3 \text{ as distinct prime factors} \end{align}$$Table 1 shows the numbers between 27838 and 30000.

Table 1: permalink

Here is the full list of the 146 numbers between 27838 and 40000 (permalink):

27838, 27956, 28126, 28215, 28258, 28329, 28340, 28424, 28458, 28614, 28728, 28768, 28782, 28809, 28826, 28985, 29029, 29222, 29260, 29295, 29337, 29393, 29512, 29640, 29667, 29678, 29835, 29848, 30039, 30184, 30240, 30264, 30305, 30381, 30504, 30566, 30760, 30780, 30814, 30888, 30914, 30943, 30956, 30996, 31008, 31027, 31160, 31174, 31283, 31331, 31392, 31416, 31465, 31496, 31529, 31806, 31816, 32103, 32130, 32131, 32298, 32376, 32395, 32589, 32604, 32718, 32802, 32984, 33015, 33176, 33292, 33345, 33383, 33440, 33480, 33495, 33497, 33528, 33572, 33592, 33836, 33885, 33915, 34008, 34162, 34276, 34293, 34317, 34440, 34452, 34573, 34580, 34605, 34782, 34884, 35061, 35074, 35112, 35340, 35343, 35424, 35464, 35530, 35752, 35805, 35910, 35948, 35960, 36366, 36423, 36666, 36828, 36859, 36860, 36890, 36920, 37060, 37128, 37417, 37638, 37719, 37730, 37758, 37772, 37961, 38038, 38152, 38285, 38340, 38368, 38408, 38610, 38745, 38760, 38874, 39032, 39121, 39219, 39270, 39370, 39458, 39501, 39520, 39556, 39576, 39729

If we consider the sum of the \( \textbf{proper} \) divisors of a number together with the totient, we find that only 104 numbers qualify in the range from 1 up to 40000. These are (permalink):

42, 78, 90, 93, 135, 198, 216, 219, 259, 270, 273, 360, 364, 403, 438, 679, 723, 738, 793, 988, 1080, 1299, 1333, 1446, 1683, 1722, 1793, 1818, 1924, 2009, 2044, 2263, 2295, 2623, 2743, 2754, 2970, 3135, 3157, 3162, 3258, 3420, 3589, 3796, 3960, 4284, 4320, 4440, 4453, 4564, 4905, 5187, 5824, 5983, 5995, 6893, 6918, 7320, 7373, 7380, 7392, 7783, 7980, 8928, 8987, 9504, 9720, 9943, 10864, 10920, 11023, 11538, 11653, 11904, 14233, 15613, 15813, 16764, 17593, 18019, 20202, 22625, 24199, 24339, 24613, 25275, 25324, 25792, 27133, 28243, 28564, 30240, 30303, 30623, 31408, 31992, 32283, 32284, 34300, 34393, 34933, 36421, 36720, 39283

Not all the prime factors of the sum of proper divisors begin with 2 as can be seen in Table 2 that shows the details for numbers between 28000 and 40000:

Table 2: permalink