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.

(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 | |
---|---|---|---|---|

1 | 1 | 8 | 672 | |

2 | 6 | 9 | 1638 | |

3 | 28 | 10 | 6200 | |

5 | 140 | 11 | 2970 | |

496 | 13 | 105664 | ||

6 | 270 | 33550336 | ||

7 | 8128 |

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.

[2] G. L. Cohen, Numbers whose positive divisors have small integral harmonic mean, Math Comp.

[3] M. Garcia, On numbers with integral harmonic mean, Amer. Math. Monthly

[4] T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math Comp.

[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.

[8] O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly

[9] C. Pomerance, Abstract 709-A5, Notices Amer. Math. Soc.

[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.

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