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We study the formulas for binomial sums of harmonic numbers of higher order ∑k=0nHk(r)nk(1−q)kqn−k=Hn(r)−∑j=1nDr(n,j)qjj. Recently, Mneimneh proved that D1(n,j)=1. In this paper, we find several different expressions of Dr(n,j) for r≥1.

It is well known that the prime number theorem can be phrased as the statement that the least common multiple (lcm) of the first n natural numbers is asymptotic to the exponential of n. Suitable weaker bounds of this lcm already suffice to deduce certain striking properties of primes such as the existence of a prime between n and 2n for sufficiently large n. In this note we prove in an elementary manner that the lcm of the first n natural numbers is bigger than when n is bigger than 6.