For a positive integer $\ n\ $ , define $$f(n)=|\sum_{j=1}^n (-1)^j\cdot j!|=n!-(n-1)!+(n-2)!-(n-3)!\pm \cdots$$
Is $\ f(n)\ $ squarefree except for $\ n=27\ $ in which case $\ 127^2\ $ is a factor ?
- for $\ n\le 60\ $ , only $\ f(27)\ $ is non-squarefree.
- There is no prime factor $\ p\le 10^6\ $ dividing $\ f(n)\ $ for some $\ n\le 1\ 000\ $ except the mentioned case.
- There is no prime factor $\ p\le 10^5\ $ dividing $\ f(n)\ $ for some $\ n\le 10^4\ $ except the mentioned case.