## (Ore's) Harmonic Numbers

#### Slides

• JANT 8 (Sep. 5, 2002)
• Seminar at University of Technology, Sydney (Sep. 20, 2004)

#### Overview

Let n be a positive integer, and tau(n),sigma(n) denote the number and the sum of the positive divisors of n, respectively. If sigma(n)=2n, then we call n perfect. In 1948, Ore [8] introduced the concept of harmonic numbers, which is extended one of perfect numbers. If the harmonic mean of positive divisors of n,

H(n)=(n tau(n))/(sigma(n))

is an integer, then we call n harmonic. Ore proved that every perfect number is harmonic, and conjectured that no nontrivial odd harmonic numbers exist (we call 1 the trivial harmonic number). If Ore's Conjecture is true, then no odd perfect numbers exist. For Ore's Conjecture, the facts below ((1)...(4)) are known. Ore proved that
(1)　a prime power is not harmonic,
and that
(2)　a squarefree integer which is greater than 6 is not harmonic.
In 1954, Garcia [3] proved that
(3)　an integer which is congruent to 3 modulo 4 is not harmonic.
In 1973, Pomerance [9] announced that he proved the following fact (in 1992, Callan [1] proved the same fact).
(4)　If n is a product of two prime powers, and is harmonic, then n is an even perfect number.
By the way, in 1957, Kanold [7] showed the following fact.
(5)　For any positive integer c, there exist only finitely many numbers n satisfying H(n)=c.
In UPINT2 [6,B2], Guy wrote: Does an integer n with H(n)=4,12,16,18,20,22 or 23 exists? In 1993, Cohen [2] found all harmonic numbers n with H(n) <= 13 (the table below), and concluded that no integers n satisfy H(n)=4 or 12.

H(n)n H(n)n
11 8672
26 91638
328 106200
5140 112970
496 13105664
6270 33550336
78128

Shibata and I [4] found all harmonic numbers n with H(n) <= 300, and concluded that no integers satisfy H(n)=16,18,20,22 or 23. Cohen [2] also listed all harmonic numbers n up to 2 times 10^9. Recently, Okeya and I [5] found all harmonic numbers n with H(n) <= 1200 (list) and all harmonic numbers less than 10^14 (list). There exist no nontrivial odd harmonic numbers in these lists, so we concluded that the following facts hold.
(6)　If n is a nontrivial odd harmonic number, then H(n) > 1200.
(7)　If n is a nontrivial odd harmonic number, then n > 10^14.
Sorli [10] found all harmonic seeds (primitive harmonic numbers) less than 10^15, so it follows that
(8)　If n is a nontrivial odd harmonic number, then n > 10^15.

#### All harmonic numbers less than 10^14 (Reference [5])

Computer searches give the
list of all harmonic numbers n with H(n)<=1200, and the list of all harmonic numbers n with H(n)^4.55>n. Since 1200^4.55>10^14, the union of these lists contain all harmonic numbers less than 10^14. Here is the list.

#### References

[1] D. Callan, Solution to Problem 6616, Amer. Math. Monthly 99 (1992), 783-789.
[2] G. L. Cohen, Numbers whose positive divisors have small integral harmonic mean, Math Comp. 66 (1997), 883-891.
[3] M. Garcia, On numbers with integral harmonic mean, Amer. Math. Monthly 61 (1954), 89-96.
[4] T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math Comp. 73 (2004), 475-491.
[5] T. Goto and K. Okeya, All harmonic numbers less than 10^14, Japan J. Indust. Appl. Math., to appear.
[6] R. K. Guy, Unsolved Problems in Number Theory, second edition, Springer-Verlag, New York, 1994.
[7] H. J. Kanold, Uber das harmonische Mittel der Teiler einer naturlichen Zahl, Math. Ann. 133 (1957), 371-374.
[8] O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly 55 (1948), 615-619.
[9] C. Pomerance, Abstract 709-A5, Notices Amer. Math. Soc. 20 (1973) A-648.
[10] R. M. Sorli, Algorithms in the Study of Multiperfect and Odd Perfect Numbers, PhD thesis, University of Technology, Sydney, 2003.

By Takeshi Goto, at Tokyo University of Science.