Let Hn be the n-th harmonic number and let vn be its denominator. It is known that vn is even for every integer . In this paper, we study the properties of Hn and prove that for any integer n, vn = en(1+o(1)). In addition, we obtain some results of the logarithmic density of harmonic numbers.
Authors:Hui-Zhou Gao, Qi Yang, Xiao-Yan Yan, Zhu-Jun Wang, Ji-Li Feng, Xia Yang, Sheng-Li Gao, Lei Feng, Xu Cheng, Chao Jia and Ke-Wu Yang
In an effort to probe the reaction of antibiotic hydrolysis catalyzed by B3 metallo-β-lactamase (MβL), the thermodynamic parameters of penicillin G hydrolysis catalyzed by MβL L1 from Stenotrophomonas maltophilia were determined by microcalorimetric method. The values of activation free energy ΔG≠θ are 88.26, 89.44, 90.49, and 91.57 kJ mol−1 at 293.15, 298.15, 303.15, and 308.15 K, respectively, activation enthalpy ΔH≠θ is 24.02 kJ mol−1, activation entropy ΔS≠θ is −219.2511 J mol−1 K−1, apparent activation energy E is 26.5183 kJ mol−1, and the reaction order is 1.0. The thermodynamic parameters reveal that the penicillin G hydrolysis catalyzed by MβL L1 is an exothermic and spontaneous reaction.