A number for which the Harmonic Mean of the Divisors of , i.e.,
, is an
Integer, where is the number of Positive integral Divisors of and is the
Divisor Function. For example, the divisors of are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140, giving

so 140 is a harmonic divisor number. Harmonic divisor numbers are also called Ore Numbers. Garcia (1954) gives the 45 harmonic divisor numbers less than . The first few are 1, 6, 140, 270, 672, 1638, ... (Sloane's A007340).

For distinct Primes and , harmonic divisor numbers are equivalent to Even Perfect Numbers for numbers of the form . Mills (1972) proved that if there exists an Odd Positive harmonic divisor number , then has a prime-Power factor greater than .

Another type of number called ``harmonic'' is the Harmonic Number.

**References**

Edgar, H. M. W. ``Harmonic Numbers.'' *Amer. Math. Monthly* **99**, 783-789, 1992.

Garcia, M. ``On Numbers with Integral Harmonic Mean.'' *Amer. Math. Monthly* **61**, 89-96, 1954.

Guy, R. K. ``Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers.''
§B2 in *Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 45-53, 1994.

Mills, W. H. ``On a Conjecture of Ore.'' *Proceedings of the 1972 Number Theory Conference.* University
of Colorado, Boulder, pp. 142-146, 1972.

Ore, Ø. ``On the Averages of the Divisors of a Number.'' *Amer. Math. Monthly* **55**, 615-619, 1948.

Pomerance, C. ``On a Problem of Ore: Harmonic Numbers.'' Unpublished manuscript, 1973.

Sloane, N. J. A. Sequence
A007340/M4299
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and extended entry in
Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Zachariou, A. and Zachariou, E. ``Perfect, Semi-Perfect and Ore Numbers.'' *Bull. Soc. Math. Gréce (New Ser.)*
**13**, 12-22, 1972.

© 1996-9

1999-05-25