  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.
