(Ore's) Harmonic Numbers
Last modified : Nov. 6, 2006
JANT 8 (Sep. 5, 2002)
Seminar at University of Technology,
Sydney (Sep. 20, 2004)
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  introduced the concept of harmonic numbers,
which is extended one of perfect numbers.
If the harmonic mean of positive divisors of 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
For Ore's Conjecture, the facts below ((1)...(4)) are known.
Ore proved that
(1) a prime power is not harmonic,
(2) a squarefree integer which is greater than 6 is not harmonic.
In 1954, Garcia  proved that
(3) an integer which is congruent to 3 modulo 4 is not harmonic.
In 1973, Pomerance  announced that he proved the following fact
(in 1992, Callan  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  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  found all harmonic numbers n with H(n) <= 13 (the
table below), and concluded that no integers n satisfy H(n)=4 or 12.
Shibata and I  found all harmonic numbers n
with H(n) <= 300, and concluded that no integers satisfy H(n)=16,18,20,22 or 23.
Cohen  also listed all harmonic numbers n up to 2 times 10^9.
Recently, Okeya and I  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  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 )
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.
 D. Callan, Solution to Problem 6616, Amer. Math.
Monthly 99 (1992), 783-789.
 G. L. Cohen, Numbers whose positive divisors have
small integral harmonic mean,
66 (1997), 883-891.
 M. Garcia, On numbers with integral harmonic mean,
Amer. Math. Monthly 61 (1954), 89-96.
 T. Goto and S. Shibata, All numbers whose positive divisors
have integral harmonic mean up to 300,
Math Comp. 73
 T. Goto and K. Okeya, All harmonic numbers less than 10^14,
Japan J. Indust. Appl. Math., to appear.
 R. K. Guy, Unsolved Problems in Number Theory, second
edition, Springer-Verlag, New York, 1994.
 H. J. Kanold, Uber das harmonische Mittel der Teiler
einer naturlichen Zahl, Math. Ann. 133 (1957), 371-374.
 O. Ore, On the averages of the divisors of a number,
Amer. Math. Monthly 55 (1948), 615-619.
 C. Pomerance, Abstract 709-A5, Notices
Amer. Math. Soc. 20 (1973) A-648.
 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.
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