Divisors

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The divisors (or factors) of a positive integer

n

are the positive integers that divide

n

without leaving a remainder. For example, 4 is a divisor of 12, since 12 divided by 4 is 3 with no remainder; 5 is not a divisor of 12 because there is a remainder of 2.
The positive divisors of

n

are the zeros of the smooth (everywhere except at

x = 0

) function

dn(x) = sin 2  (π  x) + sin 2 π  n x  , 1 ≤ x ≤ n.

Divides predicate

The divides predicate

d ∣ n

is a Boolean function which evaluates to true if and only if

d

divides

n

, otherwise evaluates to false.

Divisors of n

In the number of divisors

d (n)

column of the following table, Ramanujan’s largely composite numbers (A067128), defined to be

n

such that

d (n)   ≥   d (k)

for all

1   ≤   k < n

, are shown in bold. In the sum of divisors

σ (n)

column of the following table, the highly abundant numbers (A002093), defined as

σ (n) > σ (m)

for all

1   ≤   m < n

, are shown in bold.

Divisors of

n, n   ≥   1

.

n Divisors Count σ0 (n) (d (n), τ (n)) A000005 Sum σ1(n) (σ (n)) 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

n Divisors Count σ0 (n) (d (n), τ (n)) A000005 Sum σ1(n) (σ (n)) 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}, ...}

A027750 Triangle (sort of...) read by rows in which row

n

list the divisors of

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 divisor

di, i ∈ {1, ..., τ (n)},

of

n

, we consider the number of divisors

τ (di )

of each

di

, we have Liouville’s tau generalization, named after Joseph Liouville,

τ (n) ∑   i   = 1    τ (di  ) 3  =       τ (n) ∑   i   = 1    τ (di  )   2.

In particular, if

n

is a prime power

p n  − 1

, we have the well known

n ∑   i   = 1    i 3  =       n ∑   i   = 1    i   2.

From the prime power decomposition of

n

, we can obtain the former relation from the latter.

Aliquot divisors of n

The aliquot divisors (or aliquot parts, and unfortunately often referred to as proper divisors or proper parts) of

n

are the divisors of

n

less than

n

.

Strong divisors of n

The strong divisors (or strong parts) of

n

are the divisors of

n

greater than 1 (1 being a “weak divisor,” so to speak).

Nontrivial divisors of n

The nontrivial divisors (or nontrivial parts, which are referred to as proper divisors or proper parts in some texts) of

n

are the divisors of  

n

other than 1 or

n

. Every integer is divisible by 1, hence 1 is a trivial divisor; and every integer (except  

0

) is divisible by itself. Prime numbers have only trivial divisors.

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 divisor

d

of

n

is a unitary divisor of

n

if

d

divides

n

exactly once (i.e.

d  2

does not divide

n

). For example, 3 is a unitary divisor of 12, since 9 does not divide 12. But 2 is not a unitary divisor of 12 because 4 also divides 12 evenly.

Even unitary divisors of n

(...)

Odd unitary divisors of n

(...)

Divisors of n!

(...) ^{(Elaborate: Divisors of n!.)} 

Sequences

A000005

d (n)

(also called

τ (n)

or

σ0 (n)

), the number of divisors of

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, ...}

A067128 Ramanujan’s largely composite numbers, defined to be

n

such that

d (n)   ≥   d (k)

for

k = 1

to

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, ...}

A002182 Highly composite numbers, definition (1): where

d (n)

, the number of divisors of

n

(A000005), increases to a record.

{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, ...}

A002183 Record values of

τ (n)

: number of divisors of

n

th highly composite number.

{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, ...}

A000203

σ (n) =

sum of divisors of

n

. Also called

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, ...}

A005100 Deficient numbers: numbers

n

such that

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, ...}

A005101 Abundant numbers (sum of divisors of

n

exceeds

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, ...}

A?????? Largely abundant numbers:

σ (n)   ≥   σ (m)

for all

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)}

A002093 Highly abundant numbers:

σ (n) > σ (m)

for all

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, ...}

A034885 Record values of

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, ...}

A004394 Superabundant [or super-abundant] numbers:

n

such that

σ (n) n > σ (m) m

for all

m < n, σ (n)

being the sum of the divisors of

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, ...}

A001065 Sum of proper divisors (or aliquot parts) of

n

: sum of divisors of

n

that are less than

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, ...}

A000396 Perfect numbers

n

:

n

is equal to the sum of the proper divisors of

n

.

{6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216, ...}

A034090 Numbers

n

such that sum of proper divisors of

n

exceeds that of all smaller numbers.

{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, ...}

A007955 Product of divisors of

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, ...}

A034287 Numbers

n

such that product of divisors of

n

is larger than for any number less than

n

. (Equals A067128 for the 105834 terms less than 10 150.)

{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, ...}

A007956 Product of proper divisors of

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 on

ℤ +

, the domain of positive integers, but the concept can readily be extended to other integral domains. Let’s say

D

is some domain of algebraic integers, and

n

and

d

are in that domain. If

n d ∈ D

also, then

d

is a divisor of

n

. For example, in

ℤ [ 2√  3 ]

, we see that

1 + 2√  3

is a divisor of 2 since

2 1 + 2√  3 =  − 1 + 2√  3

. But it is not a divisor of 7 since

7 1 + 2√  3 =  −   7 2 + 7 2√  3 2 ∉ ℤ [ 2√  3 ]

.

See also

• Divisor function
  • Number of divisors function
  • Sum of divisors function
• Product of divisors function