Harmonic numbers are the partial sums of the harmonic series:
As every freshman knows (?), they grow without bound: as . It is also well known that the sequence has a finite limit, called the Euler-Mascheroni constant . On top of that, Bernoulli numbers show up in the asymptotic expansion for ,
where I take the position that . The larger gets, the fewer terms are needed for given precision.
And apparently, have something to do with the Riemann hypothesis.
In their Book of Numbers, J. H. Conway and R. K. Guy briefly entertained the idea of defining “hyperharmonic numbers” by repeating the process of partial summation.
and so on. It is natural to guess that this will produce sequences of increasing complexity. But Conway and Guy immediately destroy this illusion with the formula
The base case is a neat arithmetical trick:
For , the harmonic numbers are never integers, although makes a solid effort. The reason is simple: the highest power of not exceeding appears in some and in no other term. Nothing to cancel it with.