My diurnal age today is 27027 and this factorises as follows:$$27027=3^3 \times 7 \times 11 \times 13$$Although this number, with its many factors and 32 divisors, looks as though it should be abundant, it's not. It just misses the mark because the ratio of the sum of its proper divisors to the number itself just falls short of unity:$$ \begin{align} \frac{ \sigma(27027, 1)-27027}{27027}&=\frac{53760-27027}{27027}\\ &=\frac{26733}{27027} \\ & \approx 0.989121989121989 \end{align}$$On March 24th 2023, I wrote about Balanced Numbers and 27027 is such a number because:$$27027=\overbrace{27}^{2+7=9} \cdot 0 \cdot \overbrace{27}^{2+7=9}$$However, 27027 has a greater claim to fame because it's a member of OEIS A302934:
A302934 | Highly composite deficient numbers: deficient numbers \(k\) whose number of divisors \(d(k) \gt d(m) \) for all deficient numbers \(m \lt k\). |
The table below shows a list of deficient numbers up to one million that have a record number of divisors. The ratio of the sum of proper divisors to the number is also shown (permalink).
number divisors ratio 1 1 0.000000000000000 2 2 0.500000000000000 4 3 0.750000000000000 8 4 0.875000000000000 16 5 0.937500000000000 32 6 0.968750000000000 64 7 0.984375000000000 105 8 0.828571428571429 225 9 0.791111111111111 315 12 0.980952380952381 1155 16 0.994805194805195 2475 18 0.953939393939394 4455 20 0.955555555555556 8775 24 0.978347578347578 26325 30 0.994833808167142 27027 32 0.989121989121989 63063 36 0.974025974025974 106029 40 0.971988795518207 247401 48 0.990614427589217 693693 54 0.988980716253444 829521 60 0.995464852607710 969969 64 0.995280261534132
Looking at the table, the status of 27027 as a record breaker can be seen. Deficient numbers can be ranked by their number of divisors or by how close they approach unity (or how close they approach 2 if we prefer to deal with abundancy). I've investigated the latter in a post titled Odd Deficient Numbers from April 30th 2021. Another post on deficient numbers is Gaps Between Deficient Numbers from October 30th 2020. The post Multiperfect, Hyperfect and Superperfect Numbers from July 24th 2019 is also relevant.
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