In mathematics, the n-th hyperharmonic number of order r, denoted by
[1]: 258 By definition, the hyperharmonic numbers satisfy the recurrence relation In place of the recurrences, there is a more effective formula to calculate these numbers: The hyperharmonic numbers have a strong relation to combinatorics of permutations.
[2] The above expression with binomial coefficients easily gives that for all fixed order r>=2 we have.
[3] that is, the quotient of the left and right hand side tends to 1 as n tends to infinity.
The r=1 case for the harmonic numbers is a classical result, the general one was proved in 2009 by I. Mező and A.
[4] The next relation connects the hyperharmonic numbers to the Hurwitz zeta function:[3] It is known, that the harmonic numbers are never integers except the case n=1.
István Mező proved[5] that if r=2 or r=3, these numbers are never integers except the trivial case when n=1.
He conjectured that this is always the case, namely, the hyperharmonic numbers of order r are never integers except when n=1.
This conjecture was justified for a class of parameters by R. Amrane and H.
is not integer for all r<26 and n=2,3,... Extension to high orders was made by Göral and Sertbaş.
Then, assuming the Cramér's conjecture, Note that the number of integer lattice points in
The problem was finally settled by D. C. Sertbaş who found that there are infinitely many hyperharmonic integers, albeit they are quite huge.
The smallest hyperharmonic number which is an integer found so far is[9] [10]