Analytical form of the Eratosthenes sieve

A solution of the problem for deducing the formula expressing $i+1$-st prime number $p_{i+1}$ through $p_k$, $1\leq k\leq i$, is offered. In so doing the integer table functions $\beta_k(n)$ and $\beta'_k(n)$ are introduced. Two recurrence formulas of prime $p_{i+1}$ are derived. The second formula holds true under the assumption that between the squares of two neighbouring prime numbers there is at least one prime number.