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. "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.
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Transcription
Contents
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, such as 42 and 544635 ("abundancy" 16/7). The odd "friend" may be less than the even one, as in 84729645 and 155315394 ("abundancy" 896/351).
A square number can be friendly, for instance both 693479556 (the square of 26334) and 8640 have "abundancy" 127/36 (this example is accredited to Dean Hickerson).
Status for small n
Blue numbers are proved friendly (sequence A074902 in the OEIS), darkred 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 darkred here, though they are known to be solitary. Other numbers have unknown status and are highlighted 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 is 10; it is conjectured to be solitary. If it is not, its smallest friend is at least .^{[1]}^{[2]} Small numbers with a relatively large smallest friend do exist: for instance, 24 is "friendly", with its smallest friend 91,963,648.^{[1]}^{[2]}
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 December 2018^{[update]}, 51 perfect numbers are known, the largest of which has more than 49 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.^{[3]} Although some are known to be quite large, clubs of multiply perfect numbers (excluding the perfect numbers themselves) are conjectured to be finite.
Asymptotic density
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 gdc(n, ab) = 1. For example, the "primitive" friendly pair 6 and 28 gives rise to friendly pairs 6n and 28n for all n that are congruent to 1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, or 41 modulo 42.^{[4]}
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 desity of the solitary numbers should be 0).^{[4]}. 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.
Notes
 ^ ^{a} ^{b} Cemra, Jason. "10 Solitary Check". Github/CemraJC/Solidarity.
 ^ ^{a} ^{b} OEIS sequence A074902
 ^ Flammenkamp, Achim. "The Multiply Perfect Numbers Page". Retrieved 20080420.
 ^ ^{a} ^{b} Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) pp. 6566.
References
 Weisstein, Eric W. "Friendly Number". MathWorld.
 Weisstein, Eric W. "Friendly Pair". MathWorld.
 Weisstein, Eric W. "Solitary Number". MathWorld.
 Weisstein, Eric W. "Abundancy". MathWorld.