Geodesic
Analytical form of the Eratosthenes sieve
Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 2, pp. 583-597 
Kh. A. Smirnova. Analytical form of the Eratosthenes sieve. Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 2, pp. 583-597. http://geodesic.mathdoc.fr/item/FPM_2000_6_2_a14/
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     author = {Kh. A. Smirnova},
     title = {Analytical form of the {Eratosthenes} sieve},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {583--597},
     year = {2000},
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     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2000_6_2_a14/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.