Sums involving the largest prime divisor of an integer

On sums involving reciprocals of the largest prime factor of an integer II

Exponential Sums Involving the k-th Largest Prime Factor Function

Letting P k (n) stand for the k-th largest prime factor of n ≥ 2 and given an irrational number α and a multiplicative function f such that |f (n)| = 1 for all positive integers n, we prove that n≤x f (n) exp{2πiαP k (n)} = o(x) as x → ∞.

On the largest prime divisor of an odd harmonic number

A positive integer is called a (Ore’s) harmonic number if its positive divisors have integral harmonic mean. Ore conjectured that every harmonic number greater than 1 is even. If Ore’s conjecture is true, there exist no odd perfect numbers. In this paper, we prove that every odd harmonic number greater than 1 must be divisible by a prime greater than 105.

The third largest prime divisor of an odd perfect number exceeds one hundred

Let σ(n) denote the sum of positive divisors of the natural number n. Such a number is said to be perfect if σ(n) = 2n. It is well known that a number is even and perfect if and only if it has the form 2p−1(2p − 1) where 2p − 1 is prime. It is unknown whether or not odd perfect numbers exist, although many conditions necessary for their existence have been found. For example, Cohen and Hagis ha...

The second largest prime divisor of an odd perfect number exceeds ten thousand

Let σ(n) denote the sum of positive divisors of the natural number n. Such a number is said to be perfect if σ(n) = 2n. It is well known that a number is even and perfect if and only if it has the form 2p−1(2p − 1) where 2p − 1 is prime. No odd perfect numbers are known, nor has any proof of their nonexistence ever been given. In the meantime, much work has been done in establishing conditions ...