Problem: Do there exist infinitely many positive integers $n$ such that $$\sigma ( \varphi (n)) | n$$, where $\varphi$ is the Euler function and $\sigma$ is the sum of divisors.
Source: Own.
Attempt: Call $n$ "lucky" if $\sigma ( \varphi (n)) | n$. By programming, some small lucky numbers are found: $1,2,3,6,15,28,30,255,510,744,2418$.
Now let's consider Fermat number $${F_m} = {2^{{2^m}}} + 1.$$ It's well-known that $F_m$ is prime when $m \leqslant 4$. Let $$n = {2^{{2^k}}} - 1 = \prod\limits_{m = 0}^{k - 1} {{F_m}}$$ and it's easy to check $n$ is lucky when $k \leqslant 5$. Yet I haven't found any other lucky numbers with certain modality.
Something else to mention (seem useless): $2n$ is lucky as well if $n$ is a lucky odd number. I have also thought of induction, but the prime divisor of $\varphi( n )$ is not easy to determine.
Please help.