  In this paper we study lower ramification numbers of power series tangent to
the identity that are defined over fields of positive characteristics $p$. Let
$g$ be such a series, then $g$ has a fixed point at the origin and the
corresponding lower ramification numbers of $g$ are then, up to a constant, the
degree of the first non-linear term of $p$-power iterates of $g$. The result is
a complete characterization of power series $g$ having ramification numbers of
the form ${2(1+p+\dots+p^n)}$. Furthermore, in proving said characterization we
explicitly compute the first significant terms of $g$ at its $p$th iterate.
