Harmonic prime

A harmonic prime is a prime number that divides the numerators of exactly three harmonic numbers.

Specifically, a harmonic prime p is always a factor of the numerators of the partial harmonic sums at positions p − 1, p2 − p, and p2 − 1.

For example, the numerators of the fractions given by \sum_{i=1}^{4} \frac{1}{i} , \sum_{i=1}^{20} \frac{1}{i} , and \sum_{i=1}^{24} \frac{1}{i} 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.