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In particular, we prove the existence of the density of the primes $\\wp$ for which $d_{1, \\wp} (\\psi)$ is fixed. For $r = 2$, we also study the second elementary divisor (the exponent) of the reduction of $\\psi$ modulo $\\wp$ and prove that, on average, it has a large norm. Our work is motivated by the study of J.-P. 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