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Let $ L $ be a finite dimensional nilpotent Lie algebra and $ d $ be theminimal number generators for $ L/Z(L). $ It is known that $ \dim L/Z(L)=d \dimL^{2}-t(L)$ for an integer $ t(L)\geq 0. $ In this paper, we classify allfinite dimensional nilpotent Lie algebras $ L $ when $ t(L)\in \lbrace 0, 1, 2\rbrace.$ We find also a construction, which shows that there exist Liealgebras of arbitrary $ t(L). $

Characterizing nilpotent Lie algebras that satisfy on converse of the Schur's theorem