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

Key Points

• This technical note aims to derive a formula for counting Mersenne primes using the kappa_c function.
• Introduced the arithmetic function kappa_c(n) = prod_{d|n} (d+c) applied to Mersenne sequence M_p = 2^p - 1.
• Defined a normalized quantity a_c(M_p) to assess the primality of M_p.
• Developed the exact formula MP(P) which computes the count of Mersenne primes for given P.
• The formula MP(P) exactly counts Mersenne primes for all c >= 0.
• The normalized quantity a_c(M_p) equals 1 when M_p is prime and remains between 0 and 1 otherwise.
• The formula is valid for every c, providing a consistent method to identify Mersenne primes.