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We investigate Einstein theories of gravity, coupled to a scalar field \vphiand point-like matter, which are characterized by a scalar field-dependentmatter coupling function e^{H(\vphi)}. We show that under mild constraints onthe form of the potential for the scalar field, there are a broad class ofEinstein-like gravity models -characterized by the asymptotic behavior of H-which allow for a non-Newtonian weak-field limit with the gravitationalpotential behaving for large distances as ln r. The Newtonian term GM/r appearsonly as sub-leading. We point out that this behavior is also shared by gravitymodels described by f(R) Lagrangians. The relevance of our results for thebuilding of infrared modified theories of gravity and for modified Newtoniandynamics is also discussed.

General models of Einstein gravity with a non-Newtonian weak-field limit