Consecutive prime numerators of harmonic numbers?

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 $n>8$?

• What sort of question is this? I think it isn't trivial. Where is the conjecture from, just looking on the numbers or are there reasons why it should be right? – user302982 Feb 4 '16 at 19:40
• Just for reference, the first values of $h_n$ are listed here: oeis.org/A001008 – Thomas Andrews Feb 4 '16 at 19:49
• I edited the question to include some more values for $n$, more than oeis.org, which for some reason only includes values up to $3572$ – vrugtehagel Feb 4 '16 at 20:35
• @vrugtehagel : what do you already know about the numerators of the harmonic numbers ? – reuns Feb 4 '16 at 20:40
• @vrugtehagel Did you look in oeis.org/A056903/b056903.txt ? They call it a "b-file" and it goes up to $h_{97} = 78128$. – Mr. Brooks Feb 4 '16 at 22:01