Primes

Legendre's Formula¶

\(V_p(n!) = \sum\limits_{i=1}^\infty \lfloor \dfrac{n}{p^i} \rfloor\)

Number of divisors¶

\(n = \prod\limits_{i=1}^k p_i^{e_i}\)

\(d(n) = \prod\limits_{i=1}^k (e_i+1)\)

Sum of divisors¶

\(\sum_d (n) = \prod\limits_{i=1}^k \dfrac{p_i^{e_i+1} -1}{p_i-1}\)

Product of divisors¶

\(\prod_d (n) = n^\frac{d(n)}{2}\)