查看“︁友誼數”︁的源代码

{{unsolved|數學|存在無限多個正整數，其[[因數函數]]除以自身的比值均相等嗎？}}

在[[數論]]中，'''友誼數'''是指二個正[[整數]]m和n滿足σ(''m'')/m = σ(''n'')/''n''的關係，其中σ(n)是[[因數函數]]，則稱它們是朋友，此二個整數互為友誼數。

例如(1+2+4+5+8+10+16+20+40+80)/80 = (1+2+4+5+8+10+20+25+40+50+100+200)/200 = 93/40，因此[[80]]和[[200]]都是友誼數。

友誼數為[[传递关系]]，若m和n為友誼數，n和p為友誼數，則m和p必為友誼數。

所有的已知的友誼數有[[6]], [[12]], [[24]], [[28]], [[30]], ...（{{oeis|A074902}}，按σ(''n'')/''n''相同的組對排列：{{oeis|A050973}}、{{oeis|A050973}}）

確定不是友誼數的數即為'''孤獨數'''。但有些數尚未能證明它是否為孤獨數，例如10。

==範例==
另一個例子：30和140形成友誼數的一對，因為30和140滿足以下的等式<ref>{{cite web|url=https://tomrocksmaths.com/2023/05/10/numbers-with-cool-names-amicable-sociable-friendly/ |title=Numbers with Cool Names: Amicable, Sociable, Friendly|date=10 May 2023 |access-date=26 July 2023}}</ref>
<math> \dfrac{\sigma(30)}{30} = \dfrac{1+2+3+5+6+10+15+30}{30} =\dfrac{72}{30} = \dfrac{12}{5}</math>
:<math> \dfrac{\sigma(140)}{140} = \dfrac{1+2+4+5+7+10+14+20+28+35+70+140}{140} = \dfrac{336}{140} = \dfrac{12}{5}.</math>。

數字2480、6200、40640也是該俱樂部成員，因為它們各自的豐度等於12/5。

作為奇數的友誼數，請考慮135和819（友誼數比例16/9），也有一奇一偶的友誼數，例如：42和544635（友誼數比例16/7），奇數的朋友也可能小於偶數，例如：84729645和155315394（友誼數比例896/351）或6517665、14705145、1119251474478和2746713837618（友誼數比例64/27）。

[[平方數]]可以是友誼數，例如：693479556（26334的平方）和8640、52416的友誼數比例都是127/36，[[立方數]]也可以是友誼數，例如：3375（15的立方）和6975的友誼數比例都是416/225。

==較小整數的狀況==
在下表中，{{blue|藍色背景數字}}證明是友誼數，{{red|紅色背景數字}}證明是孤獨數，如果n和σ(''n'')互質則不標顏色，其他未知狀況用{{yellow|黃色背景數字}}標示。

<div style=display:inline-table>
{| class="wikitable"
|-
! <math>n</math> !! <math>\sigma(n)</math> !! <math>\frac{\sigma(n)}{n}</math>
|-
| 1 || 1 || 1
|-
| 2 || 3 || 3/2
|-
| 3 || 4 || 4/3
|-
| 4 || 7 || 7/4
|-
| 5 || 6 || 6/5
|- style="background: blue; color: white;"
| 6 || 12 || 2
|-
| 7 || 8 || 8/7
|-
| 8 || 15 || 15/8
|-
| 9 || 13 || 13/9
|- style="background: yellow; color: black;"
| 10 || 18 || 9/5
|-
| 11 || 12 || 12/11
|- style="background: blue; color: white;"
| 12 || 28 || 7/3
|-
| 13 || 14 || 14/13
|- style="background: yellow; color: black;"
| 14 || 24 || 12/7
|- style="background: yellow; color: black;"
| 15 || 24 || 8/5
|-
| 16 || 31 || 31/16
|-
| 17 || 18 || 18/17
|- style="background: red; color: white;"
| 18 || 39 || 13/6
|-
| 19 || 20 || 20/19
|- style="background: yellow; color: black;"
| 20 || 42 || 21/10
|-
| 21 || 32 || 32/21
|- style="background: yellow; color: black;"
| 22 || 36 || 18/11
|-
| 23 || 24 || 24/23
|- style="background: blue; color: white;"
| 24 || 60 || 5/2
|-
| 25 || 31 || 31/25
|- style="background: yellow; color: black;"
| 26 || 42 || 21/13
|-
| 27 || 40 || 40/27
|- style="background: blue; color: white;"
| 28 || 56 || 2
|-
| 29 || 30 || 30/29
|- style="background: blue; color: white;"
| 30 || 72 || 12/5
|-
| 31 || 32 || 32/31
|-
| 32 || 63 || 63/32
|- style="background: yellow; color: black;"
| 33 || 48 || 16/11
|- style="background: yellow; color: black;"
| 34 || 54 || 27/17
|-
| 35 || 48 || 48/35
|-
| 36 || 91 || 91/36
|}
</div>
<div style=display:inline-table>
{| class="wikitable"
|-
! <math>n</math> !! <math>\sigma(n)</math> !! <math>\frac{\sigma(n)}{n}</math>
|-
| 37 || 38 || 38/37
|- style="background: yellow; color: black;"
| 38 || 60 || 30/19
|-
| 39 || 56 || 56/39
|- style="background: blue; color: white;"
| 40 || 90 || 9/4
|-
| 41 || 42 || 42/41
|- style="background: blue; color: white;"
| 42 || 96 || 16/7
|-
| 43 || 44 || 44/43
|- style="background: yellow; color: black;"
| 44 || 84 || 21/11
|- style="background: red; color: white;"
| 45 || 78 || 26/15
|- style="background: yellow; color: black;"
| 46 || 72 || 36/23
|-
| 47 || 48 || 48/47
|- style="background: red; color: white;"
| 48 || 124 || 31/12
|-
| 49 || 57 || 57/49
|-
| 50 || 93 || 93/50
|- style="background: yellow; color: black;"
| 51 || 72 || 24/17
|- style="background: red; color: white;"
| 52 || 98 || 49/26
|-
| 53 || 54 || 54/53
|- style="background: yellow; color: black;"
| 54 || 120 || 20/9
|-
| 55 || 72 || 72/55
|- style="background: blue; color: white;"
| 56 || 120 || 15/7
|-
| 57 || 80 || 80/57
|- style="background: yellow; color: black;"
| 58 || 90 || 45/29
|-
| 59 || 60 || 60/59
|- style="background: blue; color: white;"
| 60 || 168 || 14/5
|-
| 61 || 62 || 62/61
|- style="background: yellow; color: black;"
| 62 || 96 || 48/31
|-
| 63 || 104 || 104/63
|-
| 64 || 127 || 127/64
|-
| 65 || 84 || 84/65
|- style="background: blue; color: white;"
| 66 || 144 || 24/11
|-
| 67 || 68 || 68/67
|- style="background: yellow; color: black;"
| 68 || 126 || 63/34
|- style="background: yellow; color: black;"
| 69 || 96 || 32/23
|- style="background: yellow; color: black;"
| 70 || 144 || 72/35
|-
| 71 || 72 || 72/71
|- style="background: yellow; color: black;"
| 72 || 195 || 65/24
|}
</div>
<div style=display:inline-table>
{| class="wikitable"
|-
! <math>n</math> !! <math>\sigma(n)</math> !! <math>\frac{\sigma(n)}{n}</math>
|-
| 73 || 74 || 74/73
|- style="background: yellow; color: black;"
| 74 || 114 || 57/37
|-
| 75 || 124 || 124/75
|- style="background: yellow; color: black;"
| 76 || 140 || 35/19
|-
| 77 || 96 || 96/77
|- style="background: blue; color: white;"
| 78 || 168 || 28/13
|-
| 79 || 80 || 80/79
|- style="background: blue; color: white;"
| 80 || 186 || 93/40
|-
| 81 || 121 || 121/81
|- style="background: yellow; color: black;"
| 82 || 126 || 63/41
|-
| 83 || 84 || 84/83
|- style="background: blue; color: white;"
| 84 || 224 || 8/3
|-
| 85 || 108 || 108/85
|- style="background: yellow; color: black;"
| 86 || 132 || 66/43
|- style="background: yellow; color: black;"
| 87 || 120 || 40/29
|- style="background: yellow; color: black;"
| 88 || 180 || 45/22
|-
| 89 || 90 || 90/89
|- style="background: yellow; color: black;"
| 90 || 234 || 13/5
|- style="background: yellow; color: black;"
| 91 || 112 || 16/13
|- style="background: yellow; color: black;"
| 92 || 168 || 42/23
|-
| 93 || 128 || 128/93
|- style="background: yellow; color: black;"
| 94 || 144 || 72/47
|- style="background: yellow; color: black;"
| 95 || 120 || 24/19
|- style="background: blue; color: white;"
| 96 || 252 || 21/8
|-
| 97 || 98 || 98/97
|-
| 98 || 171 || 171/98
|- style="background: yellow; color: black;"
| 99 || 156 || 52/33
|-
| 100 || 217 || 217/100
|-
| 101 || 102 || 102/101
|- style="background: blue; color: white;"
| 102 || 216 || 36/17
|-
| 103 || 104 || 104/103
|- style="background: yellow; color: black;"
| 104 || 210 || 105/52
|- style="background: yellow; color: black;"
| 105 || 192 || 64/35
|- style="background: yellow; color: black;"
| 106 || 162 || 81/53
|-
| 107 || 108 || 108/107
|- style="background: blue; color: white;"
| 108 || 280 || 70/27
|}
</div>
<div style=display:inline-table>
{| class="wikitable"
|-
! <math>n</math> !! <math>\sigma(n)</math> !! <math>\frac{\sigma(n)}{n}</math>
|-
| 109 || 110 || 110/109
|- style="background: yellow; color: black;"
| 110 || 216 || 108/55
|-
| 111 || 152 || 152/111
|- style="background: yellow; color: black;"
| 112 || 248 || 31/14
|-
| 113 || 114 || 114/113
|- style="background: blue; color: white;"
| 114 || 240 || 40/19
|-
| 115 || 144 || 144/115
|- style="background: yellow; color: black;"
| 116 || 210 || 105/58
|- style="background: yellow; color: black;"
| 117 || 182 || 14/9
|- style="background: yellow; color: black;"
| 118 || 180 || 90/59
|-
| 119 || 144 || 144/119
|- style="background: blue; color: white;"
| 120 || 360 || 3
|-
| 121 || 133 || 133/121
|- style="background: yellow; color: black;"
| 122 || 186 || 93/61
|- style="background: yellow; color: black;"
| 123 || 168 || 56/41
|- style="background: yellow; color: black;"
| 124 || 224 || 56/31
|-
| 125 || 156 || 156/125
|- style="background: yellow; color: black;"
| 126 || 312 || 52/21
|-
| 127 || 128 || 128/127
|-
| 128 || 255 || 255/128
|-
| 129 || 176 || 176/129
|- style="background: yellow; color: black;"
| 130 || 252 || 126/65
|-
| 131 || 132 || 132/131
|- style="background: blue; color: white;"
| 132 || 336 || 28/11
|-
| 133 || 160 || 160/133
|- style="background: yellow; color: black;"
| 134 || 204 || 102/67
|- style="background: blue; color: white;"
| 135 || 240 || 16/9
|- style="background: red; color: white;"
| 136 || 270 || 135/68
|-
| 137 || 138 || 138/137
|- style="background: blue; color: white;"
| 138 || 288 || 48/23
|-
| 139 || 140 || 140/139
|- style="background: blue; color: white;"
| 140 || 336 || 12/5
|- style="background: yellow; color: black;"
| 141 || 192 || 64/47
|- style="background: yellow; color: black;"
| 142 || 216 || 108/71
|-
| 143 || 168 || 168/143
|-
| 144 || 403 || 403/144
|}
</div>
<div style=display:inline-table>
{| class="wikitable"
|-
! <math>n</math> !! <math>\sigma(n)</math> !! <math>\frac{\sigma(n)}{n}</math>
|-style="background: yellow; color: black;"
| 145 || 180 || 36/29
|- style="background: yellow; color: black;"
| 146 || 222 || 111/73
|-style="background: yellow; color: black;"
| 147 || 228 || 76/49
|- style="background: red; color: white;"
| 148 || 266 || 133/74
|-
| 149 || 150 || 150/149
|- style="background: blue; color: white;"
| 150 || 372 || 62/25
|-
| 151 || 152 || 152/151
|- style="background: yellow; color: black;"
| 152 || 300 || 75/38
|- style="background: yellow; color: black;"
| 153 || 234 || 26/17
|- style="background: yellow; color: black;"
| 154 || 288 || 144/77
|-
| 155 || 192 || 192/155
|- style="background: yellow; color: black;"
| 156 || 392 || 98/39
|-
| 157 || 158 || 158/157
|- style="background: yellow; color: black;"
| 158 || 240 || 120/79
|- style="background: yellow; color: black;"
| 159 || 216 || 72/53
|- style="background: red; color: white;"
| 160 || 378 || 189/80
|-
| 161 || 192 || 192/161
|- style="background: red; color: white;"
| 162 || 363 || 121/54
|-
| 163 || 164 || 164/163
|-style="background: yellow; color: black;"
| 164 || 294 || 147/82
|-style="background: yellow; color: black;"
| 165 || 288 || 96/55
|- style="background: yellow; color: black;"
| 166 || 252 || 126/83
|-
| 167 || 168 || 168/167
|- style="background: blue; color: white;"
| 168 || 480 || 20/7
|-
| 169 || 183 || 183/169
|- style="background: yellow; color: black;"
| 170 || 324 || 162/85
|- 
| 171 || 260 || 260/171
|- style="background: yellow; color: black;"
| 172 || 308 || 77/43
|-
| 173 || 174 || 174/173
|- style="background: blue; color: white;"
| 174 || 360 || 60/29
|-
| 175 || 248 || 248/175
|- style="background: red; color: white;"
| 176 || 372 || 93/44
|- style="background: yellow; color: black;"
| 177 || 240 || 80/59
|- style="background: yellow; color: black;"
| 178 || 270 || 135/89
|-
| 179 || 180 || 180/179
|-style="background: yellow; color: black;"
| 180 || 546 || 91/30
|}
</div>

==大的友誼數群==
若三個或三個以上的正整數，其[[因數函數]]除以自身的比值相等，則這些正整數形成友誼數群（{{lang|en|friendly number club}}）。換言之，友誼數群是友誼數關係的[[等價類]]。目前還不知道是否有由無限多個正整數組成的友誼數群。[[完全數]]的因數函數為自身的2倍，因此所有完全數形成一個友誼數群，推測應該會有無限多個完全數（至少會和[[梅森質數]]的個數一樣多），但尚未被證明。

== 孤獨數 ==
{{unsolved|數學|10是孤獨數嗎？}}

不與其他數組成友誼數對的正整數稱為'''孤獨數'''（{{lang|en|solitary number}}）。

所有滿足( ''n'', σ(''n'') ) = 1的''n''（{{oeis|A014567}}）都是孤獨數，因此所有[[質數]][[冪]]都是孤獨數。''n'', σ(''n'')非[[互質]]的孤獨數已知有[[18]], [[45]], [[48]], ... （{{oeis|A095739}}）。10, 14, 15, 20等數未能證明它是否孤獨數。

==参考文献==
{{reflist}}

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[[Category:除數函數]]
[[Category:整数数列]]
[[Category:数论]]
[[Category:数学中未解决的问题]]