Legendre's formula and $p$-adic analysis

In number theory, we know Legendre's formula $v_p(n!) = \sum_{k \ge 1} \lfloor \frac{n}{p^k} \rfloor$, which calculates the $p$-adic valuation of the factorial, i.e. the exponent of the greatest power of a prime $p$ that divides $n!$. There is also the second (or alternative) equality $v_p (n!) = \frac{n-s_p(n)}{p-1}$ where $s_p(n)$ is the $p$-adic weight of $n$ or the sum of digits of $n$ in base $p$. Both kinds of Legendre's formula allow us to determine valuations of the natural number, the odd factorial, binomial coefficients, Catalan numbers, and other combinatorial objects. The article examines the relationship between the $p$-adic valuation and $p$-adic weight and considers their increments. The arithmetic of the $p$-adic increments is proposed.