This site is supported by donations to The OEIS Foundation.
Divisors
From OeisWiki
(Redirected from Proper 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!).