  We introduce a nonextensive entropic measure $S_{\chi}$ that grows like
$N^{\chi}$, where $N$ is the size of the system under consideration. This kind
of nonextensivity arises in a natural way in some $N$-body systems endowed with
long-range interactions described by $r^{-\alpha}$ interparticle potentials.
The power law (weakly nonextensive) behavior exhibited by $S_{\chi}$ is
intermediate between (1) the linear (extensive) regime characterizing the
standard Boltzmann-Gibbs entropy and the (2) the exponential law (strongly
nonextensive) behavior associated with the Tsallis generalized $q$-entropies.
The functional $S_{\chi} $ is parametrized by the real number $\chi \in[1,2]$
in such a way that the standard logarithmic entropy is recovered when $\chi=1$
>. We study the mathematical properties of the new entropy, showing that the
basic requirements for a well behaved entropy functional are verified, i.e.,
$S_{\chi}$ possesses the usual properties of positivity, equiprobability,
concavity and irreversibility and verifies Khinchin axioms except the one
related to additivity since $S_{\chi}$ is nonextensive. For $1<\chi<2$, the
entropy $S_{\chi}$ becomes superadditive in the thermodynamic limit. The
present formalism is illustrated by a numerical study of the thermodynamic
scaling laws of a ferromagnetic Ising model with long-range interactions.
