Friendly number
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 index of n is the rational number σ(n) / n, in which σ denotes the sum of divisors function.
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 and 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.
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.
As another example, 30 and 140 form a friendly pair, because 30 and 140 have the same abundancy:[1] The numbers 2480, 6200 and 40640 are also members of this club, as they each have an abundancy equal to 12/5.
For an example of odd numbers being friendly, consider 135 and 819 (abundancy 16/9 (deficient)).
A square number can be friendly, for instance both 693479556 (the square of 26334) and 8640 have abundancy 127/36 (this example is credited to Dean Hickerson).
In the table below, blue numbers are proven friendly (sequence A074902 in the OEIS), red numbers are proven solitary (sequence A095739 in the OEIS), numbers n such that n and
are coprime (sequence A014567 in the OEIS) are left uncolored, though they are known to be solitary.
Other numbers have unknown status and are yellow.
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 (sequence A014567 in the OEIS).
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 is 10; it is conjectured to be solitary.
other than 10 with abundancy index 9/5 must be a square with at least six distinct prime factors, the smallest being 5.
Further, at least one of the prime factors must be congruent to 1 modulo 3 and appear with an exponent congruent to 2 modulo 6 in the prime power factorization of
In [5] the authors proposed necessary upper bounds for the second, third and fourth smallest prime divisors of friends of 10, if
are the second, third, fourth smallest prime divisors of
is the number of distinct prime divisors of
[2][3] 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.
There are clubs with more known members: in particular, those formed by multiply perfect numbers, which are numbers whose abundancy is an integer.
Although some are known to be quite large, clubs of multiply perfect numbers (excluding the perfect numbers themselves) are conjectured to be finite.
Every pair a, b of friendly numbers gives rise to a positive proportion of all natural numbers being friendly (but in different clubs), by considering pairs na, nb for multipliers n with gcd(n, ab) = 1.
[6] This shows that the natural density of the friendly numbers (if it exists) is positive.
Anderson and Hickerson proposed that the density should in fact be 1 (or equivalently that the density of the solitary numbers should be 0).
[6] According to the MathWorld article on Solitary Number (see References section below), this conjecture has not been resolved, although Pomerance thought at one point he had disproved it.
