Using standard sieve methods, we were able to rapidly compute the prime counting function for primes of the form n2 + 1 up to 260≈1018; further computations take more than 1hr to compute. The program has three parameters: a starting value and a common ratio, and a boolean that toggles a detailed view of the computational results. The algorithm itself uses a residue filter mod 4 and a set of small primes to sieve out unwanted numbers. The primality test used is the Miller-Rabin with bases (2,3,5,7,11,13) for 64-bit, giving fast and exact primality tests for long inputs. Only even n's are considered, and the counting is maintained via a cumulative count across calls to avoid recomputation. The code can be modified to analyze various relations stemming from the Bateman-Horn Conjecture: for instance, we analyzed the ratio of the prime counting function for two upper bounds with a common ratio, finding that the ratio approaches the square root of the common ratio, a finding that is supported by the conjecture. Read More