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Friendly number

A friendly number is a positive natural number that shares a certain characteristic, to be defined below, with one or more other numbers. Two numbers sharing the property form a friendly pair. Larger clubs of mutually friendly numbers also exist. A number without such friends is called solitary.

The characteristic in question is the rational number σ(n) / n, in which σ denotes the divisor function.

The numbers 1 through 5 are all solitary. The smallest friendly number is 6, forming the friendly pair (6, 28), and even the friendly triplet (6, 28, 496). 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.

The divisor function

If n is a positive natural number, σ(n) is the sum of its divisors. For example, 10 is divisible by 1, 2, 5, and 10, and so σ(10) = 1 + 2 + 5 + 10 = 18.

Kinship and friendliness

Define the "kinship" κ(n) of a positive natural number n as the rational number σ(n)/n. For example, κ(10) = 18/10 = 9/5. Numbers whose kinship equals 2 are also known as perfect numbers. The name "kinship" and notation κ(n) are not standard usage, and are introduced here solely for the ease of presentation.

Numbers are mutually friendly if they share their kinship. For example, κ(6) = κ(28) = κ(496) = 2. The numbers 6, 28 and 496 are all perfect, and therefore mutually friendly. As another example, (102, 476) is a friendly pair, since κ(102) = κ(476) = 36/17.

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

Solitary numbers

The numbers that belong to a singleton club, because no other number is friendly, are the solitary numbers. All prime numbers are known to be solitary, as are powers of prime numbers. More generally, whenever 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 – the number n is solitary. For a prime number p we have σ(n) = p + 1, which is co-prime 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 2007) 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. Currently (as of 2007) 44 perfect numbers are known, so at least one club of mutually friendly numbers contains at least 44 members, the largest of which, when written out in decimal notation, is more than 19 million digits long.

Number Theory