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Divisors
From OeisWiki
(Redirected from Aliquot divisors)
n |
n |
The positive divisors of
n |
x = 0 |
-
dn(x) = sin 2 (π x) + sin 2 π
, 1 ≤ x ≤ n.n x
Contents
- 1 Divides predicate
- 2 Divisors of n
- 3 Aliquot divisors of n
- 4 Strong divisors of n
- 5 Nontrivial divisors of n
- 6 Even divisors of n
- 7 Odd divisors of n
- 8 Unitary divisors of n
- 9 Divisors of n!
- 10 Sequences
- 11 Divisor functions in computer algebra systems
- 12 Generalization to other integral domains
- 13 See also
- 14 Notes
Divides predicate
The divides predicated ∣ n |
d |
n |
Divisors of n
In the number of divisorsd (n) |
n |
d (n) ≥ d (k) |
1 ≤ k < n |
σ (n) |
σ (n) > σ (m) |
1 ≤ m < n |
n, n ≥ 1 |
|
Divisors | Count
A000005 |
Sum
A000203 | |||
---|---|---|---|---|---|---|
1 | {1} | 1 | 1 | |||
2 | {1, 2} | 2 | 3 | |||
3 | {1, 3} | 2 | 4 | |||
4 | {1, 2, 4} | 3 | 7 | |||
5 | {1, 5} | 2 | 6 | |||
6 | {1, 2, 3, 6} | 4 | 12 | |||
7 | {1, 7} | 2 | 8 | |||
8 | {1, 2, 4, 8} | 4 | 15 | |||
9 | {1, 3, 9} | 3 | 13 | |||
10 | {1, 2, 5, 10} | 4 | 18 | |||
11 | {1, 11} | 2 | 12 | |||
12 | {1, 2, 3, 4, 6, 12} | 6 | 28 | |||
13 | {1, 13} | 2 | 14 | |||
14 | {1, 2, 7, 14} | 4 | 24 | |||
15 | {1, 3, 5, 15} | 4 | 24 | |||
16 | {1, 2, 4, 8, 16} | 5 | 31 | |||
17 | {1, 17} | 2 | 18 | |||
18 | {1, 2, 3, 6, 9, 18} | 6 | 39 | |||
19 | {1, 19} | 2 | 20 | |||
20 | {1, 2, 4, 5, 10, 20} | 6 | 42 | |||
21 | {1, 3, 7, 21} | 4 | 32 | |||
22 | {1, 2, 11, 22} | 4 | 36 | |||
23 | {1, 23} | 2 | 24 | |||
24 | {1, 2, 3, 4, 6, 8, 12, 24} | 8 | 60 | |||
25 | {1, 5, 25} | 3 | 31 | |||
26 | {1, 2, 13, 26} | 4 | 42 | |||
27 | {1, 3, 9, 27} | 4 | 40 | |||
28 | {1, 2, 4, 7, 14, 28} | 6 | 56 | |||
29 | {1, 29} | 2 | 30 | |||
30 | {1, 2, 3, 5, 6, 10, 15, 30} | 8 | 72 | |||
31 | {1, 31} | 2 | 32 | |||
32 | {1, 2, 4, 8, 16, 32} | 6 | 63 | |||
33 | {1, 3, 11, 33} | 4 | 48 | |||
34 | {1, 2, 17, 34} | 4 | 54 | |||
35 | {1, 5, 7, 35} | 4 | 48 | |||
36 | {1, 2, 3, 4, 6, 9, 12, 18, 36} | 9 | 91 | |||
37 | {1, 37} | 2 | 38 | |||
38 | {1, 2, 19, 38} | 4 | 60 | |||
39 | {1, 3, 13, 39} | 4 | 56 | |||
40 | {1, 2, 4, 5, 8, 10, 20, 40} | 8 | 90 | |||
41 | {1, 41} | 2 | 42 | |||
42 | {1, 2, 3, 6, 7, 14, 21, 42} | 8 | 96 | |||
43 | {1, 43} | 2 | 44 | |||
44 | {1, 2, 4, 11, 22, 44} | 6 | 84 | |||
45 | {1, 3, 5, 9, 15, 45} | 6 | 78 | |||
46 | {1, 2, 23, 46} | 4 | 72 | |||
47 | {1, 47} | 2 | 48 | |||
48 | {1, 2, 3, 4, 6, 8, 12, 16, 24, 48} | 10 | 124 | |||
49 | {1, 7, 49} | 3 | 57 | |||
50 | {1, 2, 5, 10, 25, 50} | 6 | 93 | |||
51 | {1, 3, 17, 51} | 4 | 72 | |||
52 | {1, 2, 4, 13, 26, 52} | 6 | 98 | |||
53 | {1, 53} | 2 | 54 | |||
54 | {1, 2, 3, 6, 9, 18, 27, 54} | 8 | 120 | |||
55 | {1, 5, 11, 55} | 4 | 72 | |||
56 | {1, 2, 4, 7, 8, 14, 28, 56} | 8 | 120 | |||
57 | {1, 3, 19, 57} | 4 | 80 | |||
58 | {1, 2, 29, 58} | 4 | 90 | |||
59 | {1, 59} | 2 | 60 | |||
60 | {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60} | 12 | 168 |
|
Divisors | Count
A000005 |
Sum
A000203 | |||
---|---|---|---|---|---|---|
61 | {1, 61} | 2 | 62 | |||
62 | {1, 2, 31, 62} | 4 | 96 | |||
63 | {1, 3, 7, 9, 21, 63} | 6 | 104 | |||
64 | {1, 2, 4, 8, 16, 32, 64} | 7 | 127 | |||
65 | {1, 5, 13, 65} | 4 | 84 | |||
66 | {1, 2, 3, 6, 11, 22, 33, 66} | 8 | 144 | |||
67 | {1, 67} | 2 | 68 | |||
68 | {1, 2, 4, 17, 34, 68} | 6 | 126 | |||
69 | {1, 3, 23, 69} | 4 | 96 | |||
70 | {1, 2, 5, 7, 10, 14, 35, 70} | 8 | 144 | |||
71 | {1, 71} | 2 | 72 | |||
72 | {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72} | 12 | 195 | |||
73 | {1, 73} | 2 | 74 | |||
74 | {1, 2, 37, 74} | 4 | 114 | |||
75 | {1, 3, 5, 15, 25, 75} | 6 | 124 | |||
76 | {1, 2, 4, 19, 38, 76} | 6 | 140 | |||
77 | {1, 7, 11, 77} | 4 | 96 | |||
78 | {1, 2, 3, 6, 13, 26, 39, 78} | 8 | 168 | |||
79 | {1, 79} | 2 | 80 | |||
80 | {1, 2, 4, 5, 8, 10, 16, 20, 40, 80} | 10 | 186 | |||
81 | {1, 3, 9, 27, 81} | 5 | 121 | |||
82 | {1, 2, 41, 82} | 4 | 126 | |||
83 | {1, 83} | 2 | 84 | |||
84 | {1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84} | 12 | 224 | |||
85 | {1, 5, 17, 85} | 4 | 108 | |||
86 | {1, 2, 43, 86} | 4 | 132 | |||
87 | {1, 3, 29, 87} | 4 | 120 | |||
88 | {1, 2, 4, 8, 11, 22, 44, 88} | 8 | 180 | |||
89 | {1, 89} | 2 | 90 | |||
90 | {1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90} | 12 | 234 | |||
91 | {1, 7, 13, 91} | 4 | 112 | |||
92 | {1, 2, 4, 23, 46, 92} | 6 | 168 | |||
93 | {1, 3, 31, 93} | 4 | 128 | |||
94 | {1, 2, 47, 94} | 4 | 144 | |||
95 | {1, 5, 19, 95} | 4 | 120 | |||
96 | {1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96} | 12 | 252 | |||
97 | {1, 97} | 2 | 98 | |||
98 | {1, 2, 7, 14, 49, 98} | 6 | 171 | |||
99 | {1, 3, 9, 11, 33, 99} | 6 | 156 | |||
100 | {1, 2, 4, 5, 10, 20, 25, 50, 100} | 9 | 217 | |||
101 | {1, 101} | 2 | 102 | |||
102 | {1, 2, 3, 6, 17, 34, 51, 102} | 8 | 216 | |||
103 | {1, 103} | 2 | 104 | |||
104 | {1, 2, 4, 8, 13, 26, 52, 104} | 8 | 210 | |||
105 | {1, 3, 5, 7, 15, 21, 35, 105} | 8 | 192 | |||
106 | {1, 2, 53, 106} | 4 | 162 | |||
107 | {1, 107} | 2 | 108 | |||
108 | {1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108} | 12 | 280 | |||
109 | {1, 109} | 2 | 110 | |||
110 | {1, 2, 5, 10, 11, 22, 55, 110} | 8 | 216 | |||
111 | {1, 3, 37, 111} | 4 | 152 | |||
112 | {1, 2, 4, 7, 8, 14, 16, 28, 56, 112} | 10 | 248 | |||
113 | {1, 113} | 2 | 114 | |||
114 | {1, 2, 3, 6, 19, 38, 57, 114} | 8 | 240 | |||
115 | {1, 5, 23, 115} | 4 | 144 | |||
116 | {1, 2, 4, 29, 58, 116} | 6 | 210 | |||
117 | {1, 3, 9, 13, 39, 117} | 6 | 182 | |||
118 | {1, 2, 59, 118} | 4 | 180 | |||
119 | {1, 7, 17, 119} | 4 | 144 | |||
120 | {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120} | 16 | 360 |
The above table gives the infinite sequence of finite sequences
-
{{1}, {1, 2}, {1, 3}, {1, 2, 4}, {1, 5}, {1, 2, 3, 6}, {1, 7}, {1, 2, 4, 8}, {1, 3, 9}, {1, 2, 5, 10}, {1, 11}, {1, 2, 3, 4, 6, 12}, {1, 13}, {1, 2, 7, 14}, {1, 3, 5, 15}, {1, 2, 4, 8, 16}, {1, 17}, {1, 2, 3, 6, 9, 18}, {1, 19}, {1, 2, 4, 5, 10, 20}, {1, 3, 7, 21}, {1, 2, 11, 22}, {1, 23},
{1, 2, 3, 4, 6, 8, 12, 24}, {1, 5, 25}, {1, 2, 13, 26}, {1, 3, 9, 27}, {1, 2, 4, 7, 14, 28}, {1, 29}, {1, 2, 3, 5, 6, 10, 15, 30}, ...}
n |
n |
- {1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 3, 4, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 6, 9, 18, 1, 19, 1, 2, 4, 5, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3, 9, 27,
1, 2, 4, 7, 14, 28, 1, 29, 1, 2, 3, 5, 6, 10, 15, 30, ...}
Liouville’s tau generalization of sum of cubes equals square of sum
If, for each divisordi, i ∈ {1, ..., τ (n)}, |
n |
τ (di ) |
di |
- τ (n)
∑ i = 1τ (n)∑ i = 1
n |
p n − 1 |
- n
∑ i = 1n∑ i = 1
n |
Aliquot divisors of n
The aliquot divisors (or aliquot parts, and unfortunately often referred to as proper divisors or proper parts) ofn |
n |
n |
Strong divisors of n
The strong divisors (or strong parts) ofn |
n |
Nontrivial divisors of n
The nontrivial divisors (or nontrivial parts, which are referred to as proper divisors or proper parts in some texts) ofn |
n |
n |
0 |
For example, the nontrivial divisors of 12 are {2, 3, 4, 6}. The number 13 does not have any nontrivial divisors.
Even divisors of n
(...)
Odd divisors of n
(...)
Unitary divisors of n
A divisord |
n |
n |
d |
n |
d 2 |
n |
Even unitary divisors of n
(...)
Odd unitary divisors of n
(...)
Divisors of n!
(...) (Elaborate: Divisors of n!.) [4]
Sequences
A000005d (n) |
τ (n) |
σ0 (n) |
n |
- {1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, ...}
n |
d (n) ≥ d (k) |
k = 1 |
n − 1 |
- {1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, ...}
d (n) |
n |
- {1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, ...}
τ (n) |
n |
- {1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 32, 36, 40, 48, 60, 64, 72, 80, 84, 90, 96, 100, 108, 120, 128, 144, 160, 168, 180, 192, 200, 216, 224, 240, 256, 288, 320, 336, ...}
σ (n) = |
n |
σ1(n) |
- {1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, ...}
n |
σ (n) < 2 n |
- {1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, ...}
n |
2 n |
- {12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, ...}
σ (n) ≥ σ (m) |
m < n |
- {Is this the same sequence as A002093 Highly abundant numbers or is the strong law of small numbers at play here? — Daniel Forgues 04:30, 23 May 2012 (UTC)}
σ (n) > σ (m) |
m < n |
- {1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 210, 216, 240, 288, 300, 336, 360, 420, 480, 504, 540, 600, 630, ...}
σ (n) |
- {1, 3, 4, 7, 12, 15, 18, 28, 31, 39, 42, 60, 72, 91, 96, 124, 168, 195, 224, 234, 252, 280, 360, 403, 480, 546, 576, 600, 744, 819, 868, 992, 1170, 1344, 1512, 1560, 1680, ...}
n |
|
m < n, σ (n) |
n |
- {1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 554400, 665280, ...}
n |
n |
n |
- {0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, ...}
n |
n |
n |
- {6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216, ...}
n |
n |
- {1, 2, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 216, 240, 288, 300, 336, 360, 420, 480, 504, 540, 600, 660, 720, 840, 960, 1008, ...}
A034091 Records for sum of proper divisors function.
- {0, 1, 3, 6, 7, 8, 16, 21, 22, 36, 42, 55, 76, 108, 123, 140, 144, 156, 172, 240, 259, 312, 366, 384, 504, 531, 568, 656, 810, 924, 1032, 1056, 1140, 1260, 1356, 1698, 2040, ...}
n |
- {1, 2, 3, 8, 5, 36, 7, 64, 27, 100, 11, 1728, 13, 196, 225, 1024, 17, 5832, 19, 8000, 441, 484, 23, 331776, 125, 676, 729, 21952, 29, 810000, 31, 32768, 1089, 1156, 1225, ...}
n |
n |
n |
- {1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, ...}
n |
- {1, 1, 1, 2, 1, 6, 1, 8, 3, 10, 1, 144, 1, 14, 15, 64, 1, 324, 1, 400, 21, 22, 1, 13824, 5, 26, 27, 784, 1, 27000, 1, 1024, 33, 34, 35, 279936, 1, 38, 39, 64000, 1, 74088, 1, ...}
A034288 Product of proper divisors is larger than for any smaller number.
- {1, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, ...}
Divisor functions in computer algebra systems
The functionality is available in PARI/GP as divisors(n)
and Divisors[n]
in Mathematica.
Generalization to other integral domains
Much of the foregoing has focused onℤ + |
D |
n |
d |
|
d |
n |
ℤ [ √ 3 ] |
1 + √ 3 |
√ 3 |
√ 3 ] |
See also
Notes
- ↑ Online plot of (sin(pi*x))^2 + (sin(pi*72/x))^2.
- ↑ Peter D. Taylor, Sum of Cubes.
- ↑ Edward Barbeau and Samer Seraj, “Sum of Cubes is Square of Sum,” arXiv:1306.5257 [math.NT], 2013.
- ↑ Needs elaboration (Divisors of n!).