Let
$$\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\frac{a}{b}$$
and let $a$ and $b$ are coprime, $h_{n}=a$.
$h_{n}$ is prime for
$$n=2,3,5,8,9,21,26,41,56,62,69,79,89,91,122,127,143,167,201,230,247,252,290,349,376,459,489,492,516,662,687,714,771,932,944,1061,1281,1352,1489,1730, 1969,2012,2116,2457,2663,2955,3083,3130,3204,3359,3494,3572,3995,4155,4231,4250,4496,4616,5069,5988,6656,6883,8067,8156,8661,9097,\ldots$$
I guess proving that there are infinitely (or finitely) many primes of the form $h_n$ is very hard. But can we prove both of the $h_n$ and $h_{n+1}$ cannot be prime for $n>8$?