Odd integers $N$ with five distinct prime factors for which $2-10\sp{-12}<\sigma (N)/N<2+10\sp{-12}$

Odd perfect numbers have at least nine distinct prime factors

An odd perfect number, N , is shown to have at least nine distinct prime factors. If 3 N then N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.

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.

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...

Integers Free of Large Prime Factors

Deene (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 (x; y) in O(yf log log x log y + 1 log log y g) oating 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 other kno...

Some Remarks on Prime Factors of Integers