Divisibility by $2$ of Stirling-like numbers

Abstract:

We give a characterization of functions of the form $f(n) = \nu (n - E)$, where $\nu ( - )$ denotes the exponent of 2, and $E$ is a $2$-adic integer. We show that it applies to the restriction to even or odd integers of the function $f(n) = \nu (a{5^n} + b{3^n} + c)$, with mild restrictions on $a,b$, and $c$. This function is closely related to divisibility of certain Stirling numbers of the second kind.