Harmonic divisor number

Positive integer whose divisors have a harmonic mean that is an integer

In mathematics, a harmonic divisor number or Ore number is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are

1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 (sequence A001599 in the OEIS).

Harmonic divisor numbers were introduced by Øystein Ore, who showed that every perfect number is a harmonic divisor number and conjectured that there are no odd harmonic divisor numbers other than 1.

Examples

The number 6 has the four divisors 1, 2, 3, and 6. Their harmonic mean is an integer:

{\frac {4}{{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{6}}}}=2.

Thus 6 is a harmonic divisor number. Similarly, the number 140 has divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Their harmonic mean is

{\frac {12}{{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{10}}+{\frac {1}{14}}+{\frac {1}{20}}+{\frac {1}{28}}+{\frac {1}{35}}+{\frac {1}{70}}+{\frac {1}{140}}}}=5.

Since 5 is an integer, 140 is a harmonic divisor number.

Factorization of the harmonic mean

The harmonic mean H(n) of the divisors of any number n can be expressed as the formula

H(n)={\frac {n\sigma _{0}(n)}{\sigma _{1}(n)}}

where σ_i(n) is the sum of ith powers of the divisors of n: σ₀ is the number of divisors, and σ₁ is the sum of divisors (Cohen 1997). All of the terms in this formula are multiplicative but not completely multiplicative. Therefore, the harmonic mean H(n) is also multiplicative. This means that, for any positive integer n, the harmonic mean H(n) can be expressed as the product of the harmonic means of the prime powers in the factorization of n.

For instance, we have

H(4)={\frac {3}{1+{\frac {1}{2}}+{\frac {1}{4}}}}={\frac {12}{7}},

H(5)={\frac {2}{1+{\frac {1}{5}}}}={\frac {5}{3}},

H(7)={\frac {2}{1+{\frac {1}{7}}}}={\frac {7}{4}},

and

H(140)=H(4\cdot 5\cdot 7)=H(4)\cdot H(5)\cdot H(7)={\frac {12}{7}}\cdot {\frac {5}{3}}\cdot {\frac {7}{4}}=5.

Harmonic divisor numbers and perfect numbers

For any integer M, as Ore observed, the product of the harmonic mean and arithmetic mean of its divisors equals M itself, as can be seen from the definitions. Therefore, M is harmonic, with harmonic mean of divisors k, if and only if the average of its divisors is the product of M with a unit fraction 1/k.

Ore showed that every perfect number is harmonic. To see this, observe that the sum of the divisors of a perfect number M is exactly 2M; therefore, the average of the divisors is M(2/τ(M)), where τ(M) denotes the number of divisors of M. For any M, τ(M) is odd if and only if M is a square number, for otherwise each divisor d of M can be paired with a different divisor M/d. But no perfect number can be a square: this follows from the known form of even perfect numbers and from the fact that odd perfect numbers (if they exist) must have a factor of the form q^α where α ≡ 1 (mod 4). Therefore, for a perfect number M, τ(M) is even and the average of the divisors is the product of M with the unit fraction 2/τ(M); thus, M is a harmonic divisor number.

Ore conjectured that no odd harmonic divisor numbers exist other than 1. If the conjecture is true, this would imply the nonexistence of odd perfect numbers.

Bounds and computer searches

W. H. Mills (unpublished; see Muskat) showed that any odd harmonic divisor number above 1 must have a prime power factor greater than 10⁷, and Cohen showed that any such number must have at least three different prime factors. Cohen & Sorli (2010) showed that there are no odd harmonic divisor numbers smaller than 10²⁴.

Cohen, Goto, and others starting with Ore himself have performed computer searches listing all small harmonic divisor numbers. From these results, lists are known of all harmonic divisor numbers up to 2 × 10⁹, and all harmonic divisor numbers for which the harmonic mean of the divisors is at most 300.

References

Bogomolny, Alexander. "An Identity Concerning Averages of Divisors of a Given Integer". Retrieved 2006-09-10.
Cohen, Graeme L. (1997). "Numbers Whose Positive Divisors Have Small Integral Harmonic Mean" (PDF). Mathematics of Computation. 66 (218): 883–891. doi:10.1090/S0025-5718-97-00819-3.
Cohen, Graeme L.; Sorli, Ronald M. (2010). "Odd harmonic numbers exceed 10²⁴". Mathematics of Computation. 79 (272): 2451. doi:10.1090/S0025-5718-10-02337-9. hdl:10453/13014. ISSN 0025-5718.
Goto, Takeshi. "(Ore's) Harmonic Numbers". Retrieved 2006-09-10.
Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. B2. ISBN 978-0-387-20860-2. Zbl 1058.11001.
Muskat, Joseph B. (1966). "On Divisors of Odd Perfect Numbers". Mathematics of Computation. 20 (93): 141–144. doi:10.2307/2004277. JSTOR 2004277.
Ore, Øystein (1948). "On the averages of the divisors of a number". American Mathematical Monthly. 55 (10): 615–619. doi:10.2307/2305616. JSTOR 2305616.
Weisstein, Eric W. "Harmonic Divisor Number". MathWorld.

Divisibility-based sets of integers

Overview

Factorization forms

Constrained divisor sums

With many divisors

Aliquot sequence-related

Base-dependent

Other sets

Classes of natural numbers

Powers and related numbers
Achilles Power of 2 Power of 3 Power of 10 Square Cube Fourth power Fifth power Sixth power Seventh power Eighth power Perfect power Powerful Prime power

Of the form a × 2^b ± 1
Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall

Other polynomial numbers
Hilbert Idoneal Leyland Loeschian Lucky numbers of Euler

Recursively defined numbers
Fibonacci Jacobsthal Leonardo Lucas Narayana Padovan Pell Perrin

Possessing a specific set of other numbers
Amenable Congruent Knödel Riesel Sierpiński

Expressible via specific sums
Nonhypotenuse Polite Practical Primary pseudoperfect Ulam Wolstenholme

Figurate numbers

2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star
non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral
non-centered	Tetrahedral Cubic Octahedral Dodecahedral Icosahedral Stella octangula
pyramidal	Square pyramidal

4-dimensional

non-centered	Pentatope Squared triangular Tesseractic

Combinatorial numbers
Bell Cake Catalan Dedekind Delannoy Euler Eulerian Fuss–Catalan Lah Lazy caterer's sequence Lobb Motzkin Narayana Ordered Bell Schröder Schröder–Hipparchus Stirling first Stirling second Telephone number Wedderburn–Etherington

Primes
Wieferich Wall–Sun–Sun Wolstenholme prime Wilson

Pseudoprimes
Carmichael number Catalan pseudoprime Elliptic pseudoprime Euler pseudoprime Euler–Jacobi pseudoprime Fermat pseudoprime Frobenius pseudoprime Lucas pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime

Arithmetic functions and dynamics

Divisor functions	Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical Primitive abundant Quasiperfect Refactorable Semiperfect Sublime Superabundant Superior highly composite Superperfect
Prime omega functions	Almost prime Semiprime
Euler's totient function	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient
Aliquot sequences	Amicable Perfect Sociable Untouchable
Primorial	Euclid Fortunate

Other prime factor or divisor related numbers
Blum Cyclic Erdős–Nicolas Erdős–Woods Friendly Giuga Harmonic divisor Jordan–Pólya Lucas–Carmichael Pronic Regular Rough Smooth Sphenic Størmer Super-Poulet Zeisel

Numeral system-dependent numbers

Arithmetic functions
and dynamics

Persistence
- Additive
- Multiplicative

Digit sum	Digit sum Digital root Self Sum-product
Digit product	Multiplicative digital root Sum-product
Coding-related	Meertens
Other	Dudeney Factorion Kaprekar Kaprekar's constant Keith Lychrel Narcissistic Perfect digit-to-digit invariant Perfect digital invariant Happy