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 n-tuple.
This article relies largely or entirely on a single source. (November 2011)
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 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. 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":[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)). There are also cases of even being "friendly" to odd, such as 42, 3472, 56896, ... (sequence A347169 in the OEIS) and 544635 ("abundancy" 16/7). The odd "friend" may be less than the even one, as in 84729645 and 155315394 ("abundancy" 896/351), or in 6517665, 14705145 and 2746713837618 ("abundancy" 64/27).
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
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.
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1
1
1
2
3
3/2
3
4
4/3
4
7
7/4
5
6
6/5
6
12
2
7
8
8/7
8
15
15/8
9
13
13/9
10
18
9/5
11
12
12/11
12
28
7/3
13
14
14/13
14
24
12/7
15
24
8/5
16
31
31/16
17
18
18/17
18
39
13/6
19
20
20/19
20
42
21/10
21
32
32/21
22
36
18/11
23
24
24/23
24
60
5/2
25
31
31/25
26
42
21/13
27
40
40/27
28
56
2
29
30
30/29
30
72
12/5
31
32
32/31
32
63
63/32
33
48
16/11
34
54
27/17
35
48
48/35
36
91
91/36
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37
38
38/37
38
60
30/19
39
56
56/39
40
90
9/4
41
42
42/41
42
96
16/7
43
44
44/43
44
84
21/11
45
78
26/15
46
72
36/23
47
48
48/47
48
124
31/12
49
57
57/49
50
93
93/50
51
72
24/17
52
98
49/26
53
54
54/53
54
120
20/9
55
72
72/55
56
120
15/7
57
80
80/57
58
90
45/29
59
60
60/59
60
168
14/5
61
62
62/61
62
96
48/31
63
104
104/63
64
127
127/64
65
84
84/65
66
144
24/11
67
68
68/67
68
126
63/34
69
96
32/23
70
144
72/35
71
72
72/71
72
195
65/24
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73
74
74/73
74
114
57/37
75
124
124/75
76
140
35/19
77
96
96/77
78
168
28/13
79
80
80/79
80
186
93/40
81
121
121/81
82
126
63/41
83
84
84/83
84
224
8/3
85
108
108/85
86
132
66/43
87
120
40/29
88
180
45/22
89
90
90/89
90
234
13/5
91
112
16/13
92
168
42/23
93
128
128/93
94
144
72/47
95
120
24/19
96
252
21/8
97
98
98/97
98
171
171/98
99
156
52/33
100
217
217/100
101
102
102/101
102
216
36/17
103
104
104/103
104
210
105/52
105
192
64/35
106
162
81/53
107
108
108/107
108
280
70/27
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109
110
110/109
110
216
108/55
111
152
152/111
112
248
31/14
113
114
114/113
114
240
40/19
115
144
144/115
116
210
105/58
117
182
14/9
118
180
90/59
119
144
144/119
120
360
3
121
133
133/121
122
186
93/61
123
168
56/41
124
224
56/31
125
156
156/125
126
312
52/21
127
128
128/127
128
255
255/128
129
176
176/129
130
252
126/65
131
132
132/131
132
336
28/11
133
160
160/133
134
204
102/67
135
240
16/9
136
270
135/68
137
138
138/137
138
288
48/23
139
140
140/139
140
336
12/5
141
192
64/47
142
216
108/71
143
168
168/143
144
403
403/144
Close
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 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. If it is not, its smallest friend is at least .[2][3] .Small numbers with a relatively large smallest friend do exist: for instance, 24 is "friendly", with its smallest friend 91,963,648.[2][3]
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 December2022[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 December2022[update], the club of "friendly" numbers with "abundancy" equal to 9 has 2130 known members.[4] 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 gcd(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.[5]
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).[5] 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.
This article uses material from the Wikipedia article Friendly_number, and is written by contributors.
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