View source for Harmonic number

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Commonly, a partial sum of the [[harmonic series]]
$$
H_n = \sum_{k=1}^n \frac{1}{k} \ .
$$
A ''generalised harmonic number'' is a partial num of the [[Zeta-function|zeta function]]
$$
H_n^{(s)} = \sum_{k=1}^n \frac{1}{k^s} \ .
$$

However, Pomerance has defined a harmonic number to be a natural number $n$ for which the [[harmonic mean]] of the divisors of $n$ is an integer; equivalently $\sigma(n)$ divides $n.d(n)$ where $\sigma(n)$ is the [[Sum of divisors|sum of the divisors]] of $n$ and $d(n)$ is the [[number of divisors]]: these are also called ''Øre numbers''.  The first seven such numbers are
$$
1,\ 6,\ 28,\ 140,\ 270,\ 496,\ 672 \ .
$$
An even [[perfect number]] is a harmonic number.

====References====
* Guy, Richard K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers". Unsolved Problems in Number Theory (2nd ed.). New York: Springer-Verlag (1994). pp. 16, 45–53 
* Milovanović, Gradimir V., Rassias, Michael Th.  (edd.) ''Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava'' Springer (2014) ISBN 149390258X
* Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, edd. Handbook of number theory I. Dordrecht: Springer-Verlag (2006).  ISBN 1-4020-4215-9.  {{ZBL|1151.11300}} 
* Sándor, Jozsef; Crstici, Borislav, edd. Handbook of number theory II. Dordrecht: Kluwer Academic (2004). ISBN 1-4020-2546-7.  {{ZBL|1079.11001}}
* Wagstaff, Samuel S.  ''The Joy of Factoring'' Student mathematical library '''68''' American Mathematical Society (2013) ISBN 1470410486