A simple arithmetic challenge

Introduction
Let n be a natural number, D be the set of the divisors of n and sigma(n) be the sum of the elements of D. A number is a Zumkeller number if and only if D can be divided into two disjoint subsets D1 and D2 such that both sums of the elements of D1 and D2 equal sigma(n)/2.

Challenge
Prove (or disprove) that out of every four consecutive Zumkeller numbers there exists at least one number k such that sigma(k)/2 is also a Zumkeller number.