A Peculiarity

Sometimes, when looking for properties of the number associated with my diurnal age, I struggle to find a property that interests me. Today was one such occasion and the number was 27857. However, I did notice a "peculiarity" as I call it and it involved both the sum of the number's digits cubed and the number's proper divisors. Firstly, the digits cubed:$$ \begin{align} 27857 &\rightarrow 2^3+7^3+8^3+5^3+7^3\\ &=1331 \end{align}$$Next, let's consider the number's proper divisors:$$ \begin{align} 27857 &\rightarrow 1, 89, 313 \text{ as proper divisors} \\ \text{sum of proper divisors }&=403 \\ &=13 \times 31 \\ &\rightarrow 1331 \text{ when concatenated} \end{align} $$

This peculiarity is quite rare and in fact up to \( \textbf{one hundred million} \), there is only one other number with this same property and that is 150809. The details are:$$ \begin{align} 150809 &\rightarrow 1^3+5^3+0^3+8^3+0^3+9^3\\ &=1367\\ \\150809 &\rightarrow 1, 239, 631 \text{ as proper divisors} \\ \text{sum of proper divisors }&=871 \\ &=13\times 67 \\ &\rightarrow 1367 \text{ when concatenated} \end{align} $$So 27857 has a very interesting property after all, even if it is rather obscure. I still find it rather amazing that I was able to check up to one hundred million using SageMath on my M1 Macbook Air.