Harmonic divisor number
Encyclopedia
In mathematics
, a harmonic divisor number, or Ore number (named after Øystein Ore
who defined it in 1948), is a positive integer whose divisors have a harmonic mean
that is an integer
. The first few harmonic divisor numbers are
For example, the harmonic divisor number 6 has the four divisors 1, 2, 3, and 6. Their harmonic mean is an integer:
The number 140 has divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Their harmonic mean is:
5 is an integer, making 140 a harmonic divisor number.
of its divisors equals M itself; as is obvious from the definition. 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 qα 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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a harmonic divisor number, or Ore number (named after Øystein Ore
Øystein Ore
Øystein Ore was a Norwegian mathematician.-Life:Ore was graduated from the University of Oslo in 1922, with a Cand.Scient. degree in mathematics. In 1924, the University of Oslo awarded him the Ph.D. for a thesis titled Zur Theorie der algebraischen Körper, supervised by Thoralf Skolem...
who defined it in 1948), is a positive integer whose divisors have a harmonic mean
Harmonic mean
In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
that is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
. The first few harmonic divisor numbers are
- 1, 6, 2828 (number)28 is the natural number following 27 and preceding 29.-In mathematics:It is a composite number, its proper divisors being 1, 2, 4, 7, and 14....
, 140140 (number)140 is the natural number following 139 and preceding 141.-In mathematics:140 is an abundant number and a harmonic divisor number...
, 270270 (number)270 is the natural number following 269 and preceding 271.-In mathematics:*270 is a harmonic divisor number*270 is the fourth number that is divisible by its average integer divisor...
, 496496 (number)Four hundred [and] ninety-six is the natural number following four hundred [and] ninety-five and preceding four hundred [and] ninety-seven.-In mathematics:...
, 672, 1638, 2970, 6200, 81288128 (number)8128 is the natural number following 8127 and preceding 8129.It is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime 127, 27 − 1, with 26 · yielding 8128...
, 8190 .
For example, the harmonic divisor number 6 has the four divisors 1, 2, 3, and 6. Their harmonic mean is an integer:
The number 140 has divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Their harmonic mean is:
5 is an integer, making 140 a harmonic divisor number.
Harmonic divisor numbers and perfect numbers
For any integer M, as Ore observed, the product of the harmonic mean and arithmetic meanArithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
of its divisors equals M itself; as is obvious from the definition. 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
Unit fraction
A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/n...
1/k.
Ore showed that every perfect number
Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
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
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
of M. For any M, τ(M) is odd if and only if M is a square number
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
, 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 qα 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 and 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.