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We introduce the notion of saturation of a nilradical with respect to the eigenvalue spectrum of a maximal torus of derivations and show the existence of a generating function and a factor sequence that completely determine the nilradical. As an application, a cohomologically rigid series of solvable Lie algebras of rank one with saturated nilradical is obtained for anyand dimensions, where the eigenvalues of the torusare given by the sequence. An algorithmic procedure to search cohomologically rigid Lie algebras is proposed, based on the imposition of certain constraints on the generating function.

Algorithmic construction of solvable rigid Lie algebras determined by generating functions