MathDB

Denote by

d(n)

the number of divisors of the positive integer

n

. A positive integer

n

is called highly divisible if

d(n) > d(m)

for all positive integers

m < n

. Two highly divisible integers

m

and

n

with

m < n

are called consecutive if there exists no highly divisible integer

s

satisfying

m < s < n

. (a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form

(a, b)

with

a\mid b

. (b) Show that for every prime number

p

there exist infinitely many positive highly divisible integers

r

such that

pr

is also highly divisible.

Problem Statement