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For $m\in$, $(m,6)=1$, it is proved the relations between the sums $$W(m,s)=\sum _{i=1,(i,m)=1}^{m-1}{i}^{-s},s\in ,$$ and Bernoulli numbers. The result supplements the known theorems of C. Leudesdorf, N. Rama Rao and others. As the application it is obtained some connections between the sums $W(m,s)$ and Agoh’s functions, Wilson quotients, the indices irregularity of Bernoulli numbers.