Let κ ≥ 2 be an integer. We show that there exist infinitely many positive integers N such that the number of κ-full integers in the interval (Nκ, (N + 1)κ) is at least (log N)1/3+ο(1). We also show that the ABC-conjecture implies that for any fixed δ > 0 and sufficiently large N, the interval (N, N + N1−(2+δ)/κ) contains at most one κ-full number.
Given an integer d ≥ 2, a d-normal number, or simply a normal number, is an irrational number whose d-ary expansion is such that any preassigned sequence, of length k ≥ 1, taken within this expansion occurs at the expected limiting frequency, namely 1/d k . Answering questions raised by Igor Shparlinski, we show that 0.P(2)P(3)P(4) . . . P(n) . . . and 0.P(2 + 1)P(3 + 1)P(5 + 1) . . . P( p + 1) . . . , where P(n) stands for the largest prime factor of n, are both normal numbers.2010 Mathematics subject classification: primary 11K16; secondary 11A41, 11N37.
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