For three positive integers a, b and n, let \(H_{a,b}(n)\) be the sum of the reciprocals of the first n terms of arithmetic progression \(\{ ak+b : k=0,1, \ldots \} \) and let \(v_{a,b} (n)\) be the denominator of \(H_{a,b}(n).\) In this paper, we prove that for two coprime positive integers a and b, (i) if p is a prime with \(p\not \mid a\), then the set of positive integers n with \(p\mid v_{a,b} (n)\) has asymptotic density one; (ii) the set of positive integers n with \(v_{a,b} (n)=v_{a,b} (n+1)\) has asymptotic density one.