New Formulas for Semi-Primes. Testing, Counting and Identification of the $n^{th}$ and next Semi-Primes

In this paper we give a new semiprimality test and we construct a new formula for ${\pi }^{(2)}(N)$, the function that counts the number of semiprimes not exceeding a given number $N$. We also present new formulas to identify the $n^{th}$ semiprime and the next semiprime to a given number. The new formulas are based on the knowledge of the primes less than or equal to the cube roots of $N:P_1,P_2\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}{P}_{\pi \left(\sqrt[3]{N}\right)}\le \sqrt[3]{N}$.