A congruence for some generalized harmonic type sums

Abstract

In 1862, Wolstenholme proved that the numerator of the [Formula: see text]th harmonic number is divisible by [Formula: see text] for any prime [Formula: see text]. A variation of this theorem was shown by Alkan and Leudesdorf. Motivated by these results, we prove a congruence modulo some odd primes for some generalized harmonic type sums.

Euler sums and non-integerness of harmonic type sums

“…We extend this notation to a rational number q = a/b ∈ Q by ν p (q) = ν p (a) − ν p (b) where a, b ∈ Z. Note that for any q 1 , q 2 ∈ Q, we have 12) and the last property is called the non-Archimedian property of the p-adic valuation. Moreover we have equality in (1.12) if…”

“…and this was generalized in [12,Proposition 2.1]. Unlike the hyperharmonic case m = 1, we do not have such a formula as given in (1.8).…”