MathDB

An integer

n\geq 2

having exactly

s

positive divisors

1=d_1<d_2<\cdots<d_s=n

is said to be good if there exists an integer

k

, with

2\leq k\leq s

, such that

d_k>1+d_1+\cdots+d_{k-1}

. An integer

n\geq 2

is said to be bad if it is not good. (a) Show that there are infinitely many bad integers. (b) Prove that, among any seven consecutive integers all greater than

2

, there are always at least four good integers. (c) Show that there are infinitely many sequences of seven consecutive good integers.

Problem Statement