Harmonic prime

A harmonic prime is a prime number that divides the numerators of exactly three harmonic numbers.
Specifically, a harmonic prime is always a factor of the numerators of the partial harmonic sums at positions,, and.
For example, the numerators of the fractions given by,, and are 25, 55835135, and 1347822955, each of which is divisible by 5.
All prime numbers greater than 5 can also be found at those three indices, but many also appear at other indices. It is conjectured that there are infinitely many harmonic primes.