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.
Read more about this topic: Friendly Number
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