Fractions

Let p_i be the i_th prime. Let P(n) be the canonical product of the first n primes this way: πp_iⁱ for i=1 to n. Let #f(n) be the number of unique reduced fractions (of the form ^p⁄_q) > 1 that can be constructed using the divisors of P(n). For example #f(2) = 2 because P(2) = 2¹ × 3² = 18 The 2 fractions are ³⁄₂ and ⁹⁄₂. What is #f(10¹⁰)? Answer format: rightmost 20 digits [My timing: < 1s]