Interesting prime factorization function divisibility problem

Question

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Is the set of all numbers which divide a specific function of their prime factors, infinite?

Let the function $f(n) =(p_1^{a+1}-1)(p_2^{b+1}-1)... $ where $n$ is an integer whose factorization can be written as $p_1^a \times p_2^b...$ Find an odd integer such that $f(n)$ is divisible by $n$.

I have no idea about how to approach this. I've made some haphazard observations, but they're not coming together. Nothing under 100 seems to be working by trail and error, but I'm guessing that's not the best approach. Could someone peer at this under a lens?

Thanks.

Answer 1

Warning: this and much more can be read in the answers to a previous question; see Gerry's comment to the original question.

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Found using brute force with Mathematica: $$ 819=3^2\cdot7\cdot13 $$ $$ (3^3-1)(7^2-1)(13^2-1)=256\cdot819 $$ This is the only solution under $10^6$.