An Innovative Approach to the Finite Sequences of Prime Numbers

An innovative approach that treats prime numbers as raw experimental data making use of experimental/computational mathematics and the approximation methods is presented in order to get advanced and more exact formulations of the canonical form = being the prime value and its counter. The use of many different functions - such as the inverse of the modified chi-square function with its three parameters , and , the function with the ad-hoc values being , the function , the function , the harmonic series and its approximation by Euler and so on - as fit functions of finite sets i.e. sequences of prime numbers leads to induction algorithms and to new relationships of the kind though within the approximations of the calculations with all the estimations better than that of the standard formulation . In such a manner, refined formulations with higher precisions are got showing that there are many ways to treat the finite sequences of prime numbers. Comparisons among the various methods are made in order to find the best formulation of a new and more refined relationship in a closed form that can be valid to find the most approximate value of a prime starting from its counter in the finite case.