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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.

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    Gerry- didn't see your comment @ the time. Thank you for the link.2012-02-02

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


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 $10^6$.

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    I should clarify the intent of the question. What specifically about 819, say, makes it exactly fit the description of the function? I speak of careful mathematical observation as opposed to computation- something that could be done without a calculator. I saw this on a calculator-free set of questions I did last week and hadn't an idea about how to do it. The odd constraint made it especially difficult, as guessing and checking is futile in this problem. I know that n cannot be square free, but what observations of the mechanics of this function yields 819 in specific?2012-02-03