Friendly number

In number theory, friendly numbers are two or more natural numbers with a common abundancy, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same abundancy form a friendly pair; n numbers with the same abundancy form a friendly n-tuple.

Being mutually friendly is an equivalence relation, and thus induces a partition of the positive naturals into clubs (equivalence classes) of mutually friendly numbers.

A number that is not part of any friendly pair is called solitary.

The abundancy of n is the rational number σ(n) / n, in which σ denotes the sum of divisors function. A number n is a friendly number if there exists m ≠ n such that σ(m) / m = σ(n) / n. Note that abundancy is not the same as abundance which is defined as σ(n) − 2n.

Abundancy may also be expressed as \sigma_{-\!1}(n) where \sigma_k denotes a divisor function with \sigma_k(n) equal to the sum of the k-th powers of the divisors of n.

The numbers 1 through 5 are all solitary. The smallest friendly number is 6, forming for example the friendly pair (6, 28) with abundancy σ(6) / 6 = (1+2+3+6) / 6 = 2, the same as σ(28) / 28 = (1+2+4+7+14+28) / 28 = 2. The shared value 2 is an integer in this case but not in many other cases. There are several unsolved problems related to the friendly numbers.

In spite of the similarity in name, there is no specific relationship between the friendly numbers and the amicable numbers or the sociable numbers, although the definitions of the latter two also involve the divisor function.

Contents

Example

As another example, (30, 140) is a friendly pair, because 30 and 140 have the same abundancy:

\tfrac{\sigma(30)}{30} = \tfrac{1%2B2%2B3%2B5%2B6%2B10%2B15%2B30}{30} =\tfrac{1}{1}%2B\tfrac{1}{2}%2B\tfrac{1}{3}%2B\tfrac{1}{5}%2B\tfrac{1}{6}%2B\tfrac{1}{10}%2B\tfrac{1}{15}%2B\tfrac{1}{30}=\tfrac{12}{5}
\tfrac{\sigma(140)}{140} = \tfrac{1%2B2%2B4%2B5%2B7%2B10%2B14%2B20%2B28%2B35%2B70%2B140}{140} = \tfrac{1}{1}%2B\tfrac{1}{2}%2B\tfrac{1}{4}%2B\tfrac{1}{5}%2B\tfrac{1}{7}%2B\tfrac{1}{10}%2B\tfrac{1}{14}%2B\tfrac{1}{20}%2B\tfrac{1}{28}%2B\tfrac{1}{35}%2B\tfrac{1}{70}%2B\tfrac{1}{140}=\tfrac{12}{5}.

The numbers 2480, 6200 and 40640 are also members of this club, as they each have an abundancy equal to 12/5.

Solitary numbers

A number that belongs to a singleton club, because no other number is friendly with it, is a solitary number. All prime numbers are known to be solitary, as are powers of prime numbers. More generally, if the numbers n and σ(n) are coprime – meaning that the greatest common divisor of these numbers is 1, so that σ(n)/n is an irreducible fraction – then the number n is solitary. For a prime number p we have σ(p) = p + 1, which is coprime with p.

No general method is known for determining whether a number is friendly or solitary. The smallest number whose classification is unknown (as of 2009) is 10; it is conjectured to be solitary; if not, its smallest friend is a fairly large number.

Large clubs

It is an open problem whether there are infinitely large clubs of mutually friendly numbers. The perfect numbers form a club, and it is conjectured that there are infinitely many perfect numbers (at least as many as there are Mersenne primes), but no proof is known. As of June 2009, 47 perfect numbers are known, the largest of which has more than 25 million digits in decimal notation. There are clubs with more known members, in particular those formed by multiply perfect numbers, which are numbers whose abundancy is an integer. As of early 2008, the club of friendly numbers with abundancy equal to 9 has 2079 known members.[1] Although some are known to be quite large, clubs of multiply perfect numbers (excluding the perfect numbers themselves) are conjectured to be finite.

Notes

  1. ^ Flammenkamp, Achim. "The Multiply Perfect Numbers Page". http://wwwhomes.uni-bielefeld.de/achim/mpn.html. Retrieved 2008-04-20. 

References