On the number of integers ≤ x whose prime factors divide n

Approximating the number of integers free of large prime factors

Define Ψ(x, y) to be the number of positive integers n ≤ x such that n has no prime divisor larger than y. We present a simple algorithm that approximates Ψ(x, y) in O(y{ log log x log y + 1 log log y }) floating point operations. This algorithm is based directly on a theorem of Hildebrand and Tenenbaum. We also present data which indicate that this algorithm is more accurate in practice than o...

Approximating the number of integers without large prime factors

Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors > y. Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ(x, y) in the range (log x)1+ < y ≤ x, where is a fixed positive number ≤ 1/2. In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate Ψ(x, y). The computational complexity of this algorithm is O( √ (log ...

Some Remarks on Prime Factors of Integers

A note on the distribution of the number of prime factors of the integers

The Chernoff-Hoeffding bounds are fundamental probabilistic tools. An elementary approach is presented to obtain a Chernoff-type upper-tail bound for the number of prime factors of a random integer in {1, 2, . . . , n}. The method illustrates tail bounds in negatively-correlated settings.

Prime factors of consecutive integers

This note contains a new algorithm for computing a function f(k) introduced by Erdős to measure the minimal gap size in the sequence of integers at least one of whose prime factors exceeds k. This algorithm enables us to show that f(k) is not monotone, verifying a conjecture of Ecklund and Eggleton.