Puzzle 1170 Perfect numbers and Ore´s harmonic numbers

Ivan Ianakev, sent the following interesting puzzle:

Introduction

1. Perfect numbers (OEIS A000396) are the numbers k, such that sigma(k) = 2k, where sigma(k) is the sum of (positive integer) divisors of k.

2. Ore’s harmonic numbers (OEIS A001599) are the numbers h, such that the harmonic mean of the (positive integer) divisors of h is a positive integer. Ore proved that the perfect numbers form a subsequence of Ore’s harmonic numbers.

3. Non-trivial divisors of k are all (positive integer) divisors of k, excluding 1 and k.

Question 1

Is it provable that the perfect numbers exhibit the following property: they are Ore’s harmonic numbers whose non-trivial divisors are not Ore’s harmonic numbers?

Question 2

There is at least one Ore’s harmonic number that is not perfect, but has the property defined in Question 1 and it is 6200. Are there other such numbers?