Let 퐹(푥) be an irreducible polynomial with integer coefficients and degree at least 2. For 푥⩾푧⩾푦⩾2, denote by 퐻퐹(푥,푦,푧) the number of integers 푛⩽푥 such that 퐹(푛) has at least one divisor d with 푦<푑⩽푧. We determine the order of magnitude of 퐻퐹(푥,푦,푧) uniformly for 푦+푦/log퐶푦<푧⩽푦2 and 푦⩽푥1−훿, showing that the order is the same as the order of 퐻(푥,푦,푧), the number of positive integers 푛⩽푥 with a divisor in (푦,푧]. Here C is an arbitrarily large constant and 훿>0 is arbitrarily small.
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