The greatest prime factor of $n$, $GPF(n)$ for $1\le j\le n$ can be represented as the number of solutions to the congruence

$\displaystyle j!^n\equiv0\bmod n$

subtracted from $n+1.$ So I can express this function as an indicator function like this

$\displaystyle GPF(n)=n+1-\sum_{j=1}^{n}\lfloor\frac{j!^n}{n}\rfloor-\lfloor\frac{j!^{n}-1}{n}\rfloor=n+1-\sum_{j=1}^{n}1_{n|j!^n}.$

My question is whether or not a congruence is known for least prime factor of $n,$ $LPF(n)$ so I can represent it as an indicator function in this way. I would prefer one that is just in $n$ and $j$ like the one I have for $GPF(n),$ but I would be interested in any known congruence. Is such a congruence known? I haven't found any research and the closest one I can create on my own is

$\displaystyle 1+\sum_{j=1}^{n-1}\lfloor\frac{j^{n-1}-1}{n}\rfloor-\lfloor\frac{j^{n-1}-2}{n}\rfloor=1+\sum_{j=1}^{n-1}1_{n|j^{n-1}-1}\approx LPF(n).$

But, it's not quite exact. Thanks for any help.


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