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Some Open Problems on Number Theory

Solitary Number
A number which does not have any friends. Solitary numbers include all primes,prime powers, and numbers for which (n,sigma(n))=1, where (a,b) is the greatest common divisor of a and b and sigma(n) is the divisor function. The first few numbers satisfying (n,sigma(n))=1 are 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, ... (Sloane'sA014567). Numbers such as 18, 45, 48, 52, 136, 148, 160, 162, 176, 192, 196, 208, 232, 244, 261, 272, 292, 296, 297, 304, 320, 352, and 369 can also be easily proved to be solitary (Hickerson 2002).

Some numbers can be proved not to be solitary by finding another integer with the same index, although sometimes the smallest such number is fairly large. For example, 24 is friendly because (24,91963648) is a friendly pair. However, there exist numbers such as n=18, 45, 48, and 52 which are solitary but for which (n,sigma(n))!=1. It is believed that 10, 14, 15, 20, 22, 26, 33, 34, 38, 44, 46, 51, 54, 58, 62, 68, 69, 70, 72, 74, 76, 82, 86, 87, 88, 90, 91, 92, 94, 95, 99, 104, 105, 106, and many others are also solitary, although a proof appears to be extremely difficult.

In 1996, Carl Pomerance told Dean Hickerson that he could prove that the solitary numbers have positive density, disproving a conjecture by Anderson and Hickerson (1977). However, this proof seems not to ever have been published (Hickerson 2002).


Friendly number

From Wikipedia, the free encyclopedia
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.

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 of n is the rational number σ(n) / n, in which σ denotes 
the sum of divisors function. A number 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) − 2n.

Abundancy may also be expressed as \sigma_{-\!1}(n) where σk denotes a divisor 
function with σk(n) 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, 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.

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.  [Refer to this arXiv article for more information.]

Open Problem:  Are all (odd) squares solitary?

See initial investigations here:

Related post over at MathOverflow here:
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