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Harmonic Divisor Number


A number n for which the harmonic mean of the divisors of n, i.e., nd(n)/sigma(n), is an integer, where d(n)=sigma_0(n) is the number of positive integer divisors of n and sigma(n)=sigma_1(n) is the divisor function. For example, the divisors of n=140 are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140, giving

d(140)=12
(1)
sigma(140)=336
(2)
(140d(140))/(sigma(140))=(140·12)/(336)=5,
(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 10^7. The first few are 1, 6, 28, 140, 270, 496, ... (OEIS A001599).

For distinct primes p and q, harmonic divisor numbers are equivalent to even perfect numbers for numbers of the form p^rq. Mills (1972) proved that if there exists an odd positive harmonic divisor number n, then n has a prime-power factor greater than 10^7.

Another type of number called "harmonic" is the harmonic number.


See also

Divisor Function, Harmonic Number

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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 Unpublished manuscript, 1973.Sloane, N. J. A. Sequence A001599/M4185 in "The On-Line Encyclopedia of Integer Sequences."Sloane, N. J. A. and Plouffe, S. Figure M4299 in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." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HarmonicDivisorNumber.html

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