In [Math. Comput. 54, No. 189, 395-411 (1990)], G. L. Cohen defined and began the study of the “infinitary divisors” of an integer. In 1948 Ø. Ore [Am. Math. Mon. 55, 615-619 (1948; Zbl 0031.10903)] considered “harmonic numbers” as a generalization of perfect numbers. The harmonic mean H(n) of the positive divisors of an integer n is given by \(H(n)=n\tau (n)/\sigma (n)\) and a number is called harmonic if H(n) is an integer. (As usual, here \(\tau (n)=\sum_{d| n}1\) and \(\sigma (n)=\sum_{d| n}d)\). Clearly all perfect numbers are harmonic. The authors consider the analogue of harmonic numbers for “infinitary divisors”. It turns out that if we write an integer y in base 2: \[ y=\sum^{\infty}_{i=0}y_ i2^ i,\quad y_ i=0\quad or\quad 1; \] and let, for p a prime, \(J(p^ y)=\sum y_ i\), then J is an additive arithmetic function, and a number is infinitary harmonic if and only if \[ 2^{J(n)}\prod_{p^ y\| n;\quad p prime}\prod_{y_ i=1}\frac{p^{2^ i}}{1+p^{2^ i}} \] is an integer.
Various theorems on infinitary harmonic numbers are proved including an analogue of Dickson’s theorem on perfect numbers and a (presumably generous) estimate of the number of infinitary harmonic numbers \(\leq x\). This latter estimate is roughly \(<x^{+\epsilon}\), while there are only 38 infinitary harmonic numbers (including 1) which are \(\leq 10^ 6\). These are all given with their prime factorizations in a table.