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We examine an arithmetical function defined by recursion relations on the sequence ${\left\{f\left(p^k\right)\right\}}_{k\in \mathbb{N}}$ and obtain sufficient condition(s) for the sequence to change sign infinitely often. As an application we give criteria for infinitely many sign changes of Chebyshev polynomials and that of sequence formed by the Fourier coefficients of a cusp form.