Let S n {S_n} denote the n n th partial sum of the harmonic series. For a given positive integer k > 1 k > 1 , there exists a unique integer n k {n_k} such that S n k − 1 > k > S n k {S_{{n_k} - 1}} > k > {S_{{n_k}}} . It has been conjectured that n k {n_k} is equal to the integer nearest e k − y {e^{k - y}} , where γ \gamma is Euler’s constant. We provide an estimate on n k {n_k} which suggests that this conjecture may have to be modified. We also propose a conjecture concerning the amount by which S n k − 1 {S_{{n_k} - 1}} and S n k {S_{{n_k}}} differ from k k .