Friendly number
This article relies largely or entirely on a single source. (November 2011)

In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same abundancy form a friendly pair; n numbers with the same abundancy form a friendly ntuple.
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. 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 where denotes a divisor function with equal to the sum of the kth 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. The shared value 2 is an integer in this case but not in many other cases. Numbers with abundancy 2 are also known as perfect numbers. 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.
Examples
As another example, 30 and 140 form a friendly pair, because 30 and 140 have the same abundancy:
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). There are also cases of even being friendly to odd, like 42 and 544635 (abundancy 16/7).
A square number can be friendly, for instance both 693479556 (the square of 26334) and 8640 have abundancy 127/36 (this example is due to Dean Hickerson).
Status for small n
Blue numbers are proved friendly (sequence A074902 in the OEIS), red numbers are proved solitary (sequence A095739 in the OEIS), numbers n such that n and are coprime (sequence A014567 in the OEIS) are not coloured here, though they are known to be solitary. Other numbers have unknown status and are yellow.
n  n  n  n  
1  1  1  37  38  38/37  73  74  74/73  109  110  110/109  
2  3  3/2  38  60  30/19  74  114  57/37  110  216  108/55  
3  4  4/3  39  56  56/39  75  124  124/75  111  152  152/111  
4  7  7/4  40  90  9/4  76  140  35/19  112  248  31/14  
5  6  6/5  41  42  42/41  77  96  96/77  113  114  114/113  
6  12  2  42  96  16/7  78  168  28/13  114  240  40/19  
7  8  8/7  43  44  44/43  79  80  80/79  115  144  144/115  
8  15  15/8  44  84  21/11  80  186  93/40  116  210  105/58  
9  13  13/9  45  78  26/15  81  121  121/81  117  182  14/9  
10  18  9/5  46  72  36/23  82  126  63/41  118  180  90/59  
11  12  12/11  47  48  48/47  83  84  84/83  119  144  144/119  
12  28  7/3  48  124  31/12  84  224  8/3  120  360  3  
13  14  14/13  49  57  57/49  85  108  108/85  121  133  133/121  
14  24  12/7  50  93  93/50  86  132  66/43  122  186  93/61  
15  24  8/5  51  72  24/17  87  120  40/29  123  168  56/41  
16  31  31/16  52  98  49/26  88  180  45/22  124  224  56/31  
17  18  18/17  53  54  54/53  89  90  90/89  125  156  156/125  
18  39  13/6  54  120  20/9  90  234  13/5  126  312  52/21  
19  20  20/19  55  72  72/55  91  112  16/13  127  128  128/127  
20  42  21/10  56  120  15/7  92  168  42/23  128  255  255/128  
21  32  32/21  57  80  80/57  93  128  128/93  129  176  176/129  
22  36  18/11  58  90  45/29  94  144  72/47  130  252  126/65  
23  24  24/23  59  60  60/59  95  120  24/19  131  132  132/131  
24  60  5/2  60  168  14/5  96  252  21/8  132  336  28/11  
25  31  31/25  61  62  62/61  97  98  98/97  133  160  160/133  
26  42  21/13  62  96  48/31  98  171  171/98  134  204  102/67  
27  40  40/27  63  104  104/63  99  156  52/33  135  240  16/9  
28  56  2  64  127  127/64  100  217  217/100  136  270  135/68  
29  30  30/29  65  84  84/65  101  102  102/101  137  138  138/137  
30  72  12/5  66  144  24/11  102  216  36/17  138  288  48/23  
31  32  32/31  67  68  68/67  103  104  104/103  139  140  140/139  
32  63  63/32  68  126  63/34  104  210  105/52  140  336  12/5  
33  48  16/11  69  96  32/23  105  192  64/35  141  192  64/47  
34  54  27/17  70  144  72/35  106  162  81/53  142  216  108/71  
35  48  48/35  71  72  72/71  107  108  108/107  143  168  168/143  
36  91  91/36  72  195  65/24  108  280  70/27  144  403  403/144 
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 (sequence A014567 in the OEIS). 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, like the status for the number 24, although 24 is friendly, its smallest friend is 91,963,648. There are no friendly numbers for 10 that are less than 2,000,000,000.^{[1]}
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 February 2018^{[update]}, 50 perfect numbers are known, the largest of which has more than 46 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 2013, the club of friendly numbers with abundancy equal to 9 has 2094 known members.^{[2]} Although some are known to be quite large, clubs of multiply perfect numbers (excluding the perfect numbers themselves) are conjectured to be finite.
Notes
References
 Weisstein, Eric W. "Friendly Number". MathWorld.
 Weisstein, Eric W. "Friendly Pair". MathWorld.
 Weisstein, Eric W. "Solitary Number". MathWorld.
 Weisstein, Eric W. "Abundancy". MathWorld.