Technical note: a Mersenne-prime counting formula via the kappa_c function

This technical note applies the arithmetic function kappa_c(n) = prod_{d|n} (d+c) to the Mersenne sequence M_p = 2^p - 1. A normalized quantity a_c(M_p) is introduced, equal to 1 exactly when M_p is prime and strictly between 0 and 1 otherwise. This yields the exact formulaMP(P) = floor( sum_{p <= P, p prime} ((1+c)(2^p-1+c) / kappa_c(2^p-1))^2 )valid for every c >= 0.