Harmonic prime

Template:One source A harmonic prime Template:OEIS 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, p*(p-1), and (p-1)*(p+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.