The distribution of integers with at least two divisors in a short interval
Kevin Ford, Gerald Tenenbaum
Abstract
Let H(x,y,z) be the number of integers x with a divisor in (y,z] and let H1(x,y,z) be the number of integers x with exactly one such divisor. When y and z are close, it is expected that H1(x,y,z) H(x,y,z), that is, an integer with a divisor in (y,z] usually has just one. We determine necessary and sufficient conditions on y and z so that H1(x,y,z) H(x,y,z). In doing so, we answer an open question from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223.
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