Harmonic Divisor Number
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
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 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.
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 Unpublished manuscript, 1973.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. 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.
CITE THIS AS:
Weisstein, Eric W. "Harmonic Divisor Number."
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