  The harmonic numbers are the sequence 1, 1+1/2, 1+1/2+1/3, ... Their
asymptotic difference from the sequence of the natural logarithm of the
positive integers is Euler's constant gamma. We define a family of natural
generalizations of the harmonic numbers. The jth iterated harmonic numbers are
a sequence of rational numbers that nests the previous sequences and relates in
a similar way to the sequence of the jth iterate of the natural logarithm of
positive integers. The analogues of several well-known properties of the
harmonic numbers also hold for the iterated harmonic numbers, including a
generalization of Euler's constant. We reproduce the proof that only the first
harmonic number is an integer and, providing some numeric evidence for the
cases j = 2 and j = 3, conjecture that the same result holds for all iterated
harmonic numbers. We also review another proposed generalization of harmonic
numbers.
