Let $$1+\frac12+\frac13+\cdots+\frac1n=\frac{q}{p}$$ with $(p,q)=1$ and $q$ is a prime number.
(I) Prove or disprove that the quantity of $n$ is limited.
(II) Determine all $n$ satisfying the condition.
I use the matlab and get some $n$ meeted the condition:$2,3,5,8,9,21,26,41,56,62,69,79,89,91,122,127,143,167,201,230,247,252,290,349,376,459,489,492,516,662,687,714,771,932,944,1061,1281,1352,1489,1730, 1969,2012,2116,2457,2663,2955,3083,3130,3204,3359,3494,3572,...$
As a partial answer, see Wolstenholme’s theorem: for a prime $p > 3$, the numerator of $H_{p-1}$ is divisible by $p^2$, where
$$ H_{p-1} \equiv \sum_{n=1}^{p-1} \frac{1}{n} $$