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In this article, we consider the multiscale reduction around the harmonic solution of a general discrete nonlinear Schrodinger equation (dNLSE) depending on constant coefficients. According to the values of the coefficients we can have both integrable and non-integrable dNLSEs. For all values of the coefficients entering the dNLSE, non-secularity conditions provide an integrable NLSE at the lowest order in the perturbation parameter. However at higher order in the perturbation expansion the request that the expansion is compatible with the NLSE hierarchy gives integrability conditions which are not satisfied for the non-integrable dNLSEs. [ABSTRACT FROM AUTHOR]