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We consider a parametric Neumann problem driven by a nonlinear nonhomogeneous differential operator plus an indefinite potential term. The reaction term is superlinear but does not satisfy the Ambrosetti-Rabinowitz condition. First we prove a bifurcation-type result describing in a precise way the dependence of the set of positive solutions on the parameter $\lambda>0$. We also show the existence of a smallest positive solution. Similar results hold for the negative solutions and in this case we have a biggest negative solution. Finally using the extremal constant sign solutions we produce a smooth nodal solution.