Let $k$ be a positive integer.
What is the largest possible prime factor of a squarefree positive integer $\ n\ $ with $\ \omega(n)=k\ $ (That is, it has exactly $\ k\ $ prime factors) satisfying the condition $\ \varphi(n)\mid \sigma(n)\ $ , where $\ \varphi(n)\ $ is the totient-function and $\ \sigma(n)\ $ the divisor-sum function , assuming that such a positive integer $\ n\ $ exists at all ?
- For $k=1$ , the solutions are $2$ and $3$ , hence the largest prime factor is $3$.
- For $k=2$ , the solutions are $6,14,15,35$ , hence the largest prime factor is $7$.
- For $k=3$ , the largest prime factor seems to be $19$ , for example for $n=5\cdot 11\cdot 19$
- For $k=4$ , the lagest prime factor seems to be $127$ , for example for $n=5\cdot 13\cdot 17\cdot 127$
here is a solution with $68$ prime factors , the largest being the $13$ digit prime $5135147640209$ , hence for $k=68$ we have at least this prime factor as the largest possible.
We call $\ p(k)\ $ the largest prime factor , if a solution exists (which I assume to be the case, but I have no proof). If there is no efficient way to determine $\ p(k)\ $, is there at least a good upper bound ?