Harmonic Divisor Number
A number 
for which the Harmonic Mean of the Divisors of , i.e., 
, is anInteger, where 
is the number of Positive integral Divisors of and 
is theDivisor 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 45harmonic 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 divisornumber , then has a prime-Power factor greater than .
Another type of number called ``harmonic'' is the Harmonic Number.
See also Divisor Function, 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 Unpublished manuscript, 1973.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. SequenceA007340/M4299in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and extended entry inSloane, 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.
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