# Interesting prime factorization function divisibility problem [duplicate]

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.

• There is already a topic on this. I don't know how to make links but just type "prime factorization" in the search tool. – azarel Feb 2 '12 at 5:39
• This is fairly weaker version of the problem of finding an odd perfect number (which is an open problem). – anon Feb 2 '12 at 6:00
• Gerry- didn't see your comment @ the time. Thank you for the link. – Mathling Feb 2 '12 at 14:32

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 $10^6$.