Friendly number

In number theory, a friendly number is a natural number that shares a certain characteristic called abundancy, the ratio between the sum of divisors of the number and the number itself, with one or more other numbers. Two numbers with the same abundancy form a friendly pair. Larger clubs of mutually friendly numbers also exist. A number without such friends is called solitary.

The abundancy of "n" is the rational number σ("n") / "n", in which σ denotes the divisor function (the sum of all divisors). "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") − "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.

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.

Abundancy and friendliness

Numbers are mutually friendly if they share their abundancy. For example, 6, 28 and 496 all have abundancy 2. They are all perfect numbers, and therefore mutually friendly. As another example, (30, 140) is a friendly pair, since 30 and 140 have the same abundacy::$frac\{sigma(30)\}\{30\}\; =\; frac\{1+2+3+5+6+10+15+30\}\{30\}\; =\; frac\{12\}\{5\}$:$frac\{sigma(140)\}\{140\}\; =\; frac\{1+2+4+5+7+10+14+20+28+35+70+140\}\{140\}\; =\; frac\{12\}\{5\}.$

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

olitary 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, 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 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. As of 2008, 44 perfect numbers are known, the largest of which has more than 19 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. [cite web |last=Flammenkamp |first=Achim |title=The Multiply Perfect Numbers Page|url= http://wwwhomes.uni-bielefeld.de/achim/mpn.html |accessdate=2008-04-20 ] Although some are known to be quite large, clubs of multiply perfect numbers (excluding the perfect numbers themselves) are conjectured to be finite.

In popular culture

The webcomic xkcd mentions friendly numbers and their possible extension to the complex plane, perhaps resulting in "imaginary friends". [xkcd, [http://xkcd.com/410/ "Math Paper"] .]

Notes

References

*MathWorld|urlname=FriendlyNumber|title=Friendly Number
*MathWorld|urlname=FriendlyPair|title=Friendly Pair
*MathWorld|urlname=SolitaryNumber|title=Solitary Number
*MathWorld|urlname=Abundancy|title=Abundancy