The paper deals with the reduced harmonic series generated by four primes. A formula for the sum of these convergent reduced harmonic series is derived. These sums (concretely 42 from all 12650 sums generated by four different primes smaller than 100) are computed by using the computer algebra system Maple 15 and its programming language, although the formula is valid not only for four arbitrary primes, but also for four integers. We can say that the reduced harmonic series generated by four primes (or by four integers) belong to special types of convergent infinite series, such as geometric and telescoping series, which sum can be found analytically by means of a simple formula.