MathDB

Consecutive numbers heights

Source: IberoAmerican Olympiad For University Students

July 10, 2011

number theory proposednumber theory

Problem Statement

The height of a positive integer is defined as being the fraction

\frac{s(a)}{a}

, where

s(a)

is the sum of all the positive divisors of

a

. Show that for every pair of positive integers

N,k

there is a positive integer

b

such that the height of each of

b,b+1,\cdots,b+k

is greater than

N