# Article

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Keywords:
Wolstenholme-Leudesdorf theorem; p-integer number; Bernoulli number; Wilson quotient; irregular prime number
Summary:
For $m\in$, $(m,6)=1$, it is proved the relations between the sums $W(m,s)=\sum _{i=1, (i,m)=1}^{m-1} i^{-s}\,, \quad \quad s\in \,,$ 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.
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