Technical note: prime counting formulae via the kappa_c function

This technical note presents a family of exact prime-counting formulae based on the kappa_c function. For c >= 0, the function is defined by kappa_c(n) = productd|n (d+c). By normalizing kappa_c(n) with respect to the trivial divisors 1 and n, the note introduces a quantity E_c(n) such that E_c(n)=1 exactly for prime numbers, while composite numbers contribute a bounded residual term. This leads to the exact formula pi(N) = floor( sumₙ₌₂^N ((c+1)(n+c)/kappa_c(n))^2 ). The special cases c=0 and c=1 are detailed, together with numerical examples. The note also clarifies that the formula is structural and does not, by itself, provide a fast algorithm for computing pi(N), since exact computation of kappa_c(n) requires access to the divisors of n.