Friendly number
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| Divisibility-based sets of integers |
| Form of factorization: |
| Prime number |
| Composite number |
| Powerful number |
| Square-free number |
| Achilles number |
| Constrained divisor sums: |
| Perfect number |
| Almost perfect number |
| Quasiperfect number |
| Multiply perfect number |
| Hyperperfect number |
| Superperfect number |
| Unitary perfect number |
| Semiperfect number |
| Primitive semiperfect number |
| Practical number |
| Numbers with many divisors: |
| Abundant number |
| Highly abundant number |
| Superabundant number |
| Colossally abundant number |
| Highly composite number |
| Superior highly composite number |
| Other: |
| Deficient number |
| Weird number |
| Amicable number |
| Friendly number |
| Sociable number |
| Solitary number |
| Sublime number |
| Harmonic divisor number |
| Frugal number |
| Equidigital number |
| Extravagant number |
| See also: |
| Divisor function |
| Divisor |
| Prime factor |
| Factorization |
In number theory, a friendly number is a positive natural number that shares a certain characteristic, the ratio between the sum of divisors of the number and the number itself, with one or more other numbers. Two numbers sharing the characteristic 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 sum of all divisors). n is a friendly number if there exists m ≠ n such that σ(m) / m = 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) where 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 |
[edit] 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.
[edit] 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. The name "kinship" and notation κ(n) are not standard usage, and are introduced here solely for 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, (30, 140) is a friendly pair, since κ(30) = κ(140):
Being mutually friendly is an equivalence relation, and thus induces a partition of the positive naturals into "clubs" of mutually friendly numbers.
[edit] 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(p) = 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.
[edit] 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 kinship is an integer. As of early 2008, the club of friendly numbers with kinship 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.
[edit] References
- ^ Flammenkamp, Achim. The Multiply Perfect Numbers Page. Retrieved on 2008-04-20.
