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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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marked as duplicate by Gerry Myerson, Arturo Magidin, Ross Millikan, Aryabhata, Asaf Karagila Feb 3 '12 at 10:28

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
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

1 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.


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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No kidding. Did you look at the "possible duplicate" link in my comment to see all the work that was already done on the previous incarnation of the question? –  Gerry Myerson Feb 2 '12 at 11:29
    
@Gerry No, I did not, and I should have. –  Julián Aguirre Feb 2 '12 at 12:06
    
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 contest 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? –  Mathling Feb 2 '12 at 19:34
    
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? –  Mathling Feb 3 '12 at 1:06

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