Let $\ p\ge 5\ $ be a prime number.
How can I efficiently find the smallest squarefree number $\ n\ $ with $\ \varphi(n)\mid \sigma(n)\ $ having $\ p\ $ as the smallest prime factor , where $\sigma(n)\ $ denotes the divisor-sum-function and $\ \varphi(n)\ $ the totient-function ? If it exists , let us denote it as $\ n_p\ $.
Conjectures :
- $n_p$ exists for every prime $\ p\ge 5\ $
- The smallest solution satisfies $\ \sigma(n_p)=2\varphi(n_p)\ $
We have
- $n_5=5\cdot 7$
- $n_7=7\cdot 11\cdot 17\cdot 19$
- $n_{11}=11\cdot 13\cdot 17\cdot 19\cdot 29\cdot 31$
- $n_{13}\le 13\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 41\cdot 43$
- $n_{17}\le 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 37\cdot 41\cdot 43\cdot 47\cdot 71\cdot 97\cdot 229$
- $n_{19}\le 19\cdot 23\cdot 29\cdot 31\cdot 37\cdot 41\cdot 43\cdot 53\cdot 59\cdot 71\cdot 89\cdot 97\cdot 101\cdot 103\cdot 109\cdot 127\cdot 1217$