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Let $p_i$ denote the $i^{th}$ prime number.

Find the smallest positive integer $k$ such that the product $n = p_1 \cdot p_2 \cdots p_k$ satisfies \sigma(n) > 3n.

Is there any positive integer $m < n$ satisfying \sigma(m) > 3m?


Taken From (Rosen), at the end of the chapter under challenge problems

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    Thank you Arturo, I think I understand how to proceed now. Thanks for the push in the right direction, greatly appreciate it.2010-09-04

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This should help you out: http://www.mathhelpforum.com/math-help/f7/sum-divisors-function-question-154661.html

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The first $k$ for which $n = \prod_{i \leqslant k} p_k$ and $\sigma (n) > 3n$ is $6$, here $n = 30030$. Also, $m = 240 < 30030$ is the least integer with $\sigma (m) > 3m$.