On sums of three integers with a fixed number of prime factors

on direct sums of baer modules

the notion of baer modules was defined recently

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...

On Strings of Consecutive Integers with a Distinct Number of Prime Factors

Let ω(n) be the number of distinct prime factors of n. For any positive integer k let n = nk be the smallest positive integer such that ω(n + 1), . . . , ω(n + k) are mutually distinct. In this paper, we give upper and lower bounds for nk. We study the same quantity when ω(n) is replaced by Ω(n), the total number of prime factors of n counted with repetitions. Let ω(n) and Ω(n) denote respectiv...

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