A number
for which the harmonic mean of the divisors
of ,
i.e., ,
is an integer, where is the number of positive
integer 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
(1)
(2)
(3)
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, 28, 140, 270, 496, ... (OEIS A001599).
For distinct primes and , harmonic divisor numbers are equivalent to evenperfect numbers for numbers of
the form.
Mills (1972) proved that if there exists an oddpositive harmonic divisor number , then has a prime-power factor greater
than .
Another type of number called "harmonic" is the harmonic
number.
Edgar, H. M. W. "Harmonic Numbers." Amer. Math. Monthly99, 783-789, 1992.Garcia, M. "On
Numbers with Integral Harmonic Mean." Amer. Math. Monthly61,
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. Monthly55, 615-619, 1948.Pomerance, C. "On
a Problem of Ore: Harmonic Numbers." Unpublished manuscript, 1973.Sloane,
N. J. A. Sequence A001599/M4185
in "The On-Line Encyclopedia of Integer Sequences."Sloane,
N. J. A. and Plouffe, S. Figure M4299 in 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.
for which the harmonic mean of the divisors
of 
,
i.e., 
,
is an integer, where 
is the number of positive
integer 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
. The first few are 1, 6, 28, 140, 270, 496, ... (OEIS A001599).
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 
.
