Like the entire harmonic series, the harmonic series sum_(k = 1)^∞ 1/p_k = ∞ taken over all primes p_k also diverges, as first shown by Euler in 1737 (Nagell 1951, p. 59, pp. 17 and 22; Wells 1986, p. 41; Havil 2003, pp. 28-31), although it does so very slowly. The sum exceeds 1, 2, 3, ... after 3, 59, 361139, ... (OEIS A046024) primes.

harmonic series | prime sums