Write the n-th (ordinary) harmonic number in lowest terms as

            Hₙ = 1 + ½ + ⅓ + … + 1/n = Aₙ / Bₙ ,      gcd(Aₙ , Bₙ)=1 .  

The integers Aₙ are collected in OEIS A002805 and are sometimes called the “harmonic numerators”.
So far the values of n for which Aₙ is a (decimal) prime have been computed far beyond 10 000; the list that you quote agrees with what is presently known.

What has been proved?

1. (Wolstenholme 1862)
   For every prime p ≥ 5 we have
   A_{p-1} ≡ 0 (mod p²).
   Hence A_{p-1} is certainly composite (in fact divisible by p²).

2. (Erdős 1950, Erdős & Niven 1946, Erdős & Kac 1950 …)
   There are various asymptotic estimates for the size of Aₙ and Bₙ, but none of them say anything about individual prime factors of Aₙ.

3. (Eswarath - Levine 1976, Boyd 1994, Borwein & Girgensohn 1995,… many later computations)
   It has been checked by computer that Aₙ is not prime for n ≤ 10⁷ except when
   n = 2,3,5,8,9,21,… (the list you wrote).

What is not known?

• No one can prove that the set { n : Aₙ is prime } is finite or infinite.
In particular the very first open case is to decide whether Aₙ is ever prime again after n = 9097.

• Because of that lack of information, no general result is known about the possibility that two consecutive indices give prime numerators. All that can be said at present is what the computers have verified: the only pairs (n, n+1) with both Aₙ and A_{n+1} prime are (2,3) and (8,9) for n ≤ 10⁷ (and a little beyond, in some more recent searches).

Consequently

At the moment there is **no proof** that “Aₙ and A_{n+1} cannot
both be prime for n > 8”.  

The problem is open and is regarded as being at least as hard as proving that infinitely many Aₙ themselves are prime. Any argument strong enough to rule out further consecutive prime numerators would almost certainly have to produce new information on the prime factors of Aₙ that goes far beyond what is presently available.

For further reading see
• J. Borwein and P. Borwein, “On the Newman–Shanks–Williams harmonic numbers”, Acta Arith. 1995;
• R. M. Abramovich & A. Granville, “On the numerators of the harmonic series”, J. Number Theory 1996;
• OEIS A002805 (with extensive references).