This technical note presents a family of exact prime-counting formulae based on the kappac function. For c >= 0, the function is defined by kappac (n) = product₃|₍ (d+c). By normalizing kappac (n) with respect to the trivial divisors 1 and n, the note introduces a quantity Ec (n) such that Ec (n) =1 exactly for prime numbers, while composite numbers contribute a bounded residual term. This leads to the exact formula pi (N) = floor (sum₍=₂N ( (c+1) (n+c) /kappac (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 kappac (n) requires access to the divisors of n.
Christophe POKORSKI (Sun,) studied this question.