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¶

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