Abstract Prime Number Theorem provide asymptotic estimates but do not yield exact results. This paper presents a complete, closed-form mathematical equation that exactly computes the prime-counting function π(N) for any integer N ≥ 2. Unlike existing methods which are either asymptotic approximations or recursive algorithms, our formulation is a single evaluable expression. The equation operates in two distinct modes: (1) using a sequence of known primes, or (2) using the simple sequence J₁ = 2, Jₙ = (n-1)-th odd integer ≥ 3 for n ≥ 2, with an intrinsic primality test μₙ = ⌈∏ₓ₌₁ⁿ⁻¹ (Jₙ/Jₓ - ⌊Jₙ/Jₓ⌋)⌉ where μₙ = 1 if and only if Jₙ is prime. The formula directly yields π(N) through elementary arithmetic operations without recursion, iteration, or algorithmic procedures. The implications of this formula are explored in comparison to existing prime counting functions and its potential impact on the study of prime distribution, it is an explicit sieve-theoretic expression and a self-contained rewriting. This complements classical exact prime-counting methods (Meissel–Lehmer and descendants), which are vastly more efficient for computation.