Divisor function

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The divisor function

σk (n), k ∈ ℤ,

for a positive integer

n

is defined as the sum of the

k

th powers of the divisors of

n

${\sigma }_k(n):=\sum _{d|n}{d}^k,k\in \mathbb{\mathbb{Z}}.$

In particular

• σ − 1(n)

  is the harmonic sum of divisors of

  n

  ;

• σ0(n)

  is the number of divisors of

  n

  , and is often notated

  τ (n)

  or

  d (n)

  ;

• σ1(n)

  is the sum of divisors of

  n

  and is often notated

  σ (n)

  .

Formulae for the divisor function

From the prime factorization of

n

$n=\prod _{\stackrel{i=1}{{p_i}^{{\alpha }_i}\parallel n,{\alpha }_i\ge 1}}^{\omega (n)}{p_i}^{{\alpha }_i},$

where the

pi

are the distinct prime factors of

n

and

ω (n)

is the number of distinct prime factors of

n

, we obtain the divisor function of

n

${\sigma }_k(n)=\prod _{i=1}^{\omega (n)}\sum _{j=0}^{{\alpha }_i}{p_i}^{jk}=\prod _{i=1}^{\omega (n)}(1+{p_i}^k+{p_i}^{2k}+\cdots +{p_i}^{{\alpha }_{i}k})$

since for each

pi

we can choose the exponent from 0 to

αi

to build a divisor of

n

, and which simplifies to

${\sigma }_k(n)=\{\begin{array}{ll}\prod _{i=1}^{\omega (n)}(1+{\alpha }_i)& \text{if }k=0,\\ \prod _{i=1}^{\omega (n)}\frac{{p_i}^{({\alpha }_i+1)k}-1}{{p_i}^k-1}& \text{if }k>0.\end{array}$

Generating function of the divisor function

The generating function is

$G_{\{{\sigma }_k(n)\}}(x)\equiv \sum _{n=1}^{\mathrm{\infty }}{\sigma }_k(n)x^n=?.$

Dirichlet generating function of the divisor function

The Dirichlet generating function is

$D_{\{{\sigma }_k(n)\}}(s)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{{\sigma }_k(n)}{n^s}=\zeta (s)\zeta (s-k)?$

σ₋₁(n): Harmonic sum of divisors function

For

k = −1

we get

${\sigma }_{-1}(n):=\sum _{d|n}{d}^{-1}=\sum _{d|n}\frac{1}{d}=\sum _{d|n}\frac{1}{n/d}=\sum _{d|n}\frac{d}{n}=\frac{1}{n}\sum _{d|n}d=\frac{{\sigma }_1(n)}{n},$

where

σ−1(n)

is the harmonic sum of divisors of

n

. If the harmonic sum of divisors

σ−1(n) = k

is a positive integer

k

,

n

is a k-perfect number since

σ1(n) = k n

.

σ₀(n): Number of divisors function

For

k = 0

we get the number of divisors

${\sigma }_0(n):=\sum _{d|n}{d}^0=\sum _{d|n}1=:D(n)=:d(n)=:\nu (n)=:\tau (n),$

where

τ (n)

is the number of divisors function. The notations

d (n)

,

ν (n)

, and

τ (n)

are sometimes used for

σ0(n)

, which gives the number of divisors of

n

. For

n > 0

, the number of divisors is the number of restricted partitions with parts of equal size.

s₀(n): Number of aliquot divisors function

For

k = 0

we get the number of aliquot divisors (number of divisors less than

n

)

$s_0(n):={\sigma }_0(n)-1,$

where

σ0 (n)

is the number of divisors function.

σ₁(n): Sum of divisors function

For

k = 1

we get the sum of divisors

${\sigma }_1(n):=\sum _{d|n}{d}^1=\sum _{d|n}d=:\sigma (n),$

where

σ (n)

is the sum of divisors function. The notation

σ (n)

is often used for

σ1(n)

, which gives the sum of divisors of

n

.

s₁(n): Sum of aliquot divisors function

For

k = 1

we get the sum of aliquot divisors (sum of divisors less than

n

)

$s_1(n):={\sigma }_1(n)-n,$

where

σ1 (n)

is the sum of divisors function.

Table of related formulae and values

Divisor function related formulae and values

k	Formulae σk (n) =   ω (n) ∏   i  = 1    (αi + 1), k = 0,   ω (n) ∏   i  = 1    pi k  (αi +1) − 1 pi k − 1 , k ≥ 1.	Generating function G{σk (n)}(x) =   ∞ ∑   i  = 1    i  k x i 1 − x i	Dirichlet generating function D{σk (n)}(s) = ζ (s) ζ (s − k)?	Differences σk (n) − σk (n − 1) =	Partial sums   m ∑   n  = 1    σk (n) =	Partial sums of reciprocals   m ∑   n  = 1    1 σk (n) =	Sum of reciprocals   ∞ ∑   n  = 1    1 σk (n) =
0	$\prod _{i=1}^{\omega (n)}({\alpha }_i+1)$	$\sum _{i=1}^{\mathrm{\infty }}\frac{x^i}{1-x^i}$	$(\zeta (s){)}^2$
1	$\prod _{i=1}^{\omega (n)}\frac{p_i^{{\alpha }_i+1}-1}{p_i-1}$	$\sum _{i=1}^{\mathrm{\infty }}i\frac{x^i}{1-x^i}$	$\zeta (s)\zeta (s-1)$
2	$\prod _{i=1}^{\omega (n)}\frac{p_i^{2({\alpha }_i+1)}-1}{{p_i}^2-1}$	$\sum _{i=1}^{\mathrm{\infty }}{i}^2\frac{x^i}{1-x^i}$	$\zeta (s)\zeta (s-2)$
3	$\prod _{i=1}^{\omega (n)}\frac{p_i^{3({\alpha }_i+1)}-1}{{p_i}^3-1}$	$\sum _{i=1}^{\mathrm{\infty }}{i}^3\frac{x^i}{1-x^i}$	$\zeta (s)\zeta (s-3)?$
4	$\prod _{i=1}^{\omega (n)}\frac{p_i^{4({\alpha }_i+1)}-1}{{p_i}^4-1}$	$\sum _{i=1}^{\mathrm{\infty }}{i}^4\frac{x^i}{1-x^i}$	$\zeta (s)\zeta (s-4)?$
5	$\prod _{i=1}^{\omega (n)}\frac{p_i^{5({\alpha }_i+1)}-1}{{p_i}^5-1}$	$\sum _{i=1}^{\mathrm{\infty }}{i}^5\frac{x^i}{1-x^i}$	$\zeta (s)\zeta (s-5)?$
6	$\prod _{i=1}^{\omega (n)}\frac{p_i^{6({\alpha }_i+1)}-1}{{p_i}^6-1}$	$\sum _{i=1}^{\mathrm{\infty }}{i}^6\frac{x^i}{1-x^i}$	$\zeta (s)\zeta (s-6)?$
7	$\prod _{i=1}^{\omega (n)}\frac{p_i^{7({\alpha }_i+1)}-1}{{p_i}^7-1}$	$\sum _{i=1}^{\mathrm{\infty }}{i}^7\frac{x^i}{1-x^i}$	$\zeta (s)\zeta (s-7)?$
8	$\prod _{i=1}^{\omega (n)}\frac{p_i^{8({\alpha }_i+1)}-1}{{p_i}^8-1}$	$\sum _{i=1}^{\mathrm{\infty }}{i}^8\frac{x^i}{1-x^i}$	$\zeta (s)\zeta (s-8)?$
9	$\prod _{i=1}^{\omega (n)}\frac{p_i^{9({\alpha }_i+1)}-1}{{p_i}^9-1}$	$\sum _{i=1}^{\mathrm{\infty }}{i}^9\frac{x^i}{1-x^i}$	$\zeta (s)\zeta (s-9)?$
10	$\prod _{i=1}^{\omega (n)}\frac{p_i^{10({\alpha }_i+1)}-1}{{p_i}^{10}-1}$	$\sum _{i=1}^{\mathrm{\infty }}{i}^{10}\frac{x^i}{1-x^i}$	$\zeta (s)\zeta (s-10)?$
11	$\prod _{i=1}^{\omega (n)}\frac{p_i^{11({\alpha }_i+1)}-1}{{p_i}^{11}-1}$	$\sum _{i=1}^{\mathrm{\infty }}{i}^{11}\frac{x^i}{1-x^i}$	$\zeta (s)\zeta (s-11)?$
12	$\prod _{i=1}^{\omega (n)}\frac{p_i^{12({\alpha }_i+1)}-1}{{p_i}^{12}-1}$	$\sum _{i=1}^{\mathrm{\infty }}{i}^{12}\frac{x^i}{1-x^i}$	$\zeta (s)\zeta (s-12)?$

Table of sequences

Divisor function sequences

k	σk (n), n   ≥   1.	A-number
0	{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, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, ...}	A000005  (n)
1	{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, 72, 48, 124, 57, 93, 72, ...}	A000203  (n)
2	{1, 5, 10, 21, 26, 50, 50, 85, 91, 130, 122, 210, 170, 250, 260, 341, 290, 455, 362, 546, 500, 610, 530, 850, 651, 850, 820, 1050, 842, 1300, 962, 1365, 1220, 1450, 1300, 1911, 1370, 1810, ...}	A001157  (n)
3	{1, 9, 28, 73, 126, 252, 344, 585, 757, 1134, 1332, 2044, 2198, 3096, 3528, 4681, 4914, 6813, 6860, 9198, 9632, 11988, 12168, 16380, 15751, 19782, 20440, 25112, 24390, 31752, 29792, ...}	A001158  (n)
4	{1, 17, 82, 273, 626, 1394, 2402, 4369, 6643, 10642, 14642, 22386, 28562, 40834, 51332, 69905, 83522, 112931, 130322, 170898, 196964, 248914, 279842, 358258, 391251, 485554, ...}	A001159  (n)
5	{1, 33, 244, 1057, 3126, 8052, 16808, 33825, 59293, 103158, 161052, 257908, 371294, 554664, 762744, 1082401, 1419858, 1956669, 2476100, 3304182, 4101152, 5314716, 6436344, ...}	A001160  (n)
6	{1, 65, 730, 4161, 15626, 47450, 117650, 266305, 532171, 1015690, 1771562, 3037530, 4826810, 7647250, 11406980, 17043521, 24137570, 34591115, 47045882, 65019786, 85884500, ...}	A013954  (n)
7	{1, 129, 2188, 16513, 78126, 282252, 823544, 2113665, 4785157, 10078254, 19487172, 36130444, 62748518, 106237176, 170939688, 270549121, 410338674, 617285253, 893871740, ...}	A013955  (n)
8	{1, 257, 6562, 65793, 390626, 1686434, 5764802, 16843009, 43053283, 100390882, 214358882, 431733666, 815730722, 1481554114, 2563287812, 4311810305, 6975757442, ...}	A013956  (n)
9	{1, 513, 19684, 262657, 1953126, 10097892, 40353608, 134480385, 387440173, 1001953638, 2357947692, 5170140388, 10604499374, 20701400904, 38445332184, 68853957121, ...}	A??????  (n)
10	{1, 1025, ...}	A??????  (n)
11	{1, 2049, ...}	A??????  (n)
12	{1, 4097, ...}	A??????  (n)

See also

• Numbers with relatively many divisors
• Numbers with relatively many and large divisors

• Divisor function
  • Number of divisors function (number of divisors)
  • Sum of divisors function (sum of divisors)
  • Divisorial function (divisorial, product of divisors)

• Even divisors function
  • Number of even divisors function (number of even divisors)
  • Sum of even divisors function (sum of even divisors)
  • Product of even divisors function (product of even divisors)

• Odd divisors function
  • Number of odd divisors function (number of odd divisors)
    • Number of divisors of form 4 m + 1
    • Number of divisors of form 4 m + 3
    • (number of divisors of form 4 m + 1)  −  (number of divisors of form 4 m + 3)
  • Sum of odd divisors function (sum of odd divisors)
    • Sum of divisors of form 4 m + 1
    • Sum of divisors of form 4 m + 3
    • (sum of divisors of form 4 m + 1)  −  (sum of divisors of form 4 m + 3)
  • Product of odd divisors function (product of odd divisors)
    • Product of divisors of form 4 m + 1
    • Product of divisors of form 4 m + 3
    • (product of divisors of form 4 m + 1)  −  (product of divisors of form 4 m + 3)

Arithmetic function templates

• {{divisor function}} or {{sigma k}} arithmetic function template (for

  k ∈ ℤ＼{0}

  )
• {{number of divisors}} or {{sigma 0}} or {{tau}} arithmetic function template (for

  k = 0

  )
• {{sum of divisors}} or {{sigma 1}} or {{sigma}} arithmetic function template (for

  k = 1

  )

References

• Burton, D. M. (1989). Elementary Number Theory (4th ed.). Boston, MA: Allyn and Bacon. 

• Hardy, G. H.; Wright, E. M. (1979). An Introduction to the Theory of Numbers (5th ed.). Oxford, England: Oxford University Press. pp. 354–355. 

• Knopp, K. (1951). Theory and Application of Infinite Series. London: Blackie. p. 451. 

• Ore, Ø. (1988). Number Theory and Its History. New York: Dover. 

• Titchmarsh, E. C. (1938). “On a series of Lambert type”. J. London Math. Soc. 13: pp. 248–253.