Iterated Harmonic Numbers

SummaryThe harmonic numbers are the sequence 1,1+1/2,1+1/2+1/3,…. Asymptotically, the difference between the nth harmonic number and the natural logarithm of n converges to Euler’s constant γ. We define a family of natural, iterated generalizations of the harmonic numbers. The jth iterated harmonic numbers build upon the previous sequences in a natural way, and they relate to iterated logarithms much like ordinary harmonic numbers relate to the natural logarithm. We find that the analogues of several well-known properties of the harmonic numbers also hold for the iterated harmonic numbers, including finding generalizations of Euler’s constant. For the second-order case, we compute the first six digits of this constant, γ2=0.887118…. After reviewing the proof that only the first harmonic number is an integer and providing some numeric evidence, we conjecture the same result holds for all iterated harmonic numbers. We also review another proposed extension of harmonic numbers.