Problem 2

Find all integers

n

satisfying

n\ge2

and

\frac{\sigma(n)}{p(n)-1}=n,

in which

\sigma(n)

denotes the sum of all positive divisors of

n

, and

p(n)

denotes the largest prime divisor of

n

Step 2 of 3: Force at most two prime factors, both at most 3

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k\ge3\ \Longrightarrow\ p_k-1\ge 2k-2\ge k+1,\text{ contradiction}\ \Longrightarrow\ k\le2,\ p_k\le3

Detailed analysis

For $k\ge3$ , the $k$ -th prime is at least $2k-1$ , so $p_k-1\ge2k-2\ge k+1$ , contradicting $p_k-1<k+1$ . Thus $k\le2$ , and $p_k<k+2\le4$ forces $p_k\le3$ .