A number of the form

(1) |

(2) |

(3) | |||

(4) |

The first few harmonic numbers are 1, , , , , ... (Sloane's A001008 and A002805). The Harmonic Number is never an Integer (except for ), which can be proved by using the strong triangle inequality to show that the 2-adic value of is greater than 1 for . The harmonic numbers have Odd Numerators and Even Denominators. The th harmonic number is given asymptotically by

(5) |

(6) |

(7) | |||

(8) | |||

(9) |

where is the Riemann Zeta Function. The first of these had been previously derived by de Doelder (1991), and the last by Euler (1775). These identities are corollaries of the identity

(10) |

(11) |

(12) |

Conway and Guy (1996) define the second harmonic number by

(13) |

(14) |

(15) |

(16) | |||

(17) |

plus the recurrence relation

(18) |

(19) |

(20) |

(21) |

A separate type of number sometimes also called a ``harmonic number'' is a Harmonic Divisor Number (or Ore Number).

**References**

Borwein, D. and Borwein, J. M. ``On an Intriguing Integral and Some Series Related to .''
*Proc. Amer. Math. Soc.* **123**, 1191-1198, 1995.

Conway, J. H. and Guy, R. K. *The Book of Numbers.* New York: Springer-Verlag, pp. 143 and 258-259, 1996.

de Doelder, P. J. ``On Some Series Containing
and
for Certain Values
of and .'' *J. Comp. Appl. Math.* **37**, 125-141, 1991.

Roman, S. ``The Logarithmic Binomial Formula.'' *Amer. Math. Monthly* **99**, 641-648, 1992.

Sloane, N. J. A. Sequences
A001008/M2885
and A002805/M1589
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

© 1996-9

1999-05-25