Abstract
It is conjectured that the sum
S-r(n) = Sigma(n)(k=1) k/k + r ((n)(k))
for positive integers r, n is never integral. This has been shown for r <= 22. In this note we study the problem in the "n aspect" showing that the set of n such that S-r (n) is an element of Z for some r >= 1 has asymptotic density 0. Our principal tools are some deep results on the distribution of primes in short intervals.