Three consecutive almost squares

Three Consecutive Almost Squares

Given a positive integer n, we let sfp(n) denote the squarefree part of n. We determine all positive integers n for which max{sfp(n), sfp(n+ 1), sfp(n+ 2)} ≤ 150 by relating the problem to finding integral points on elliptic curves. We also prove that there are infinitely many n for which max{sfp(n), sfp(n + 1), sfp(n + 2)} < n.

Finding Almost Squares

We study short intervals which contain an " almost square " , an integer n that can be factored as n = ab with a, b close to √ n. This is related to the problem on distribution of n 2 α (mod 1) and the problem on gaps between sums of two squares.

Set covering with almost consecutive ones property

In this paper we consider set covering problems with a coefficient matrix almost having the consecutive ones property, i.e., in many rows of the coefficient matrix, the ones appear consecutively. If this property holds for all rows it is well known that the set covering problem can be solved efficiently. For our case of almost consecutive ones we present a reformulation exploiting the consecuti...

Almost Perfect Powers in Consecutive Integers (ii)

Let k 4 be an integer. We find all integers of the form by where l 2 and the greatest prime factor of b is at most k (i.e. nearly a perfect power) such that they are also products of k consecutive integers with two terms omitted.

Squares from Products of Consecutive Integers