From the text: The author considers the von Staudt type congruences for Bernoulli numbers with arbitrary indices (the case being no exception). The theorems proved generalize well-known results due to H. S. Vandiver, L. Carlitz and others.
Theorem 1. Let be an odd prime, , and . Then or more exactly
Theorem 2. Let and . Then
where , and . In particular, for we have
or (in the usual symbolic form)
where and . denotes the Bernoulli number.