Divisor function

This article needs more work.

Please help by expanding it!

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

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

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

$n=\prod _{\stackrel {i=1}{{p_{i}}^{\alpha _{i}}\parallel \,n,\,\alpha _{i}\geq 1}}^{\omega (n)}{p_{i}}^{\alpha _{i}},\,$

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

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

Generating function of the divisor function

The generating function is

$G_{\{\sigma _{k}(n)\}}(x)\equiv \sum _{n=1}^{\infty }\sigma _{k}(n)\,x^{n}=~?.\,$

Dirichlet generating function of the divisor function

The Dirichlet generating function is

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

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	$\prod _{i=1}^{\omega (n)}(\alpha _{i}+1)\,$	$\sum _{i=1}^{\infty }{\frac {x^{i}}{1-x^{i}}}\,$	$(\zeta (s))^{2}\,$
1	$\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{\alpha _{i}+1}-1}{p_{i}-1}}\,$	$\sum _{i=1}^{\infty }i{\frac {x^{i}}{1-x^{i}}}\,$	$\zeta (s)\,\zeta (s-1)\,$
2	$\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{2(\alpha _{i}+1)}-1}{{p_{i}}^{2}-1}}\,$	$\sum _{i=1}^{\infty }i^{2}{\frac {x^{i}}{1-x^{i}}}\,$	$\zeta (s)\,\zeta (s-2)\,$
3	$\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{3(\alpha _{i}+1)}-1}{{p_{i}}^{3}-1}}\,$	$\sum _{i=1}^{\infty }i^{3}{\frac {x^{i}}{1-x^{i}}}\,$	$\zeta (s)\,\zeta (s-3)?\,$
4	$\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{4(\alpha _{i}+1)}-1}{{p_{i}}^{4}-1}}\,$	$\sum _{i=1}^{\infty }i^{4}{\frac {x^{i}}{1-x^{i}}}\,$	$\zeta (s)\,\zeta (s-4)?\,$
5	$\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{5(\alpha _{i}+1)}-1}{{p_{i}}^{5}-1}}\,$	$\sum _{i=1}^{\infty }i^{5}{\frac {x^{i}}{1-x^{i}}}\,$	$\zeta (s)\,\zeta (s-5)?\,$
6	$\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{6(\alpha _{i}+1)}-1}{{p_{i}}^{6}-1}}\,$	$\sum _{i=1}^{\infty }i^{6}{\frac {x^{i}}{1-x^{i}}}\,$	$\zeta (s)\,\zeta (s-6)?\,$
7	$\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{7(\alpha _{i}+1)}-1}{{p_{i}}^{7}-1}}\,$	$\sum _{i=1}^{\infty }i^{7}{\frac {x^{i}}{1-x^{i}}}\,$	$\zeta (s)\,\zeta (s-7)?\,$
8	$\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{8(\alpha _{i}+1)}-1}{{p_{i}}^{8}-1}}\,$	$\sum _{i=1}^{\infty }i^{8}{\frac {x^{i}}{1-x^{i}}}\,$	$\zeta (s)\,\zeta (s-8)?\,$
9	$\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{9(\alpha _{i}+1)}-1}{{p_{i}}^{9}-1}}\,$	$\sum _{i=1}^{\infty }i^{9}{\frac {x^{i}}{1-x^{i}}}\,$	$\zeta (s)\,\zeta (s-9)?\,$
10	$\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{10(\alpha _{i}+1)}-1}{{p_{i}}^{10}-1}}\,$	$\sum _{i=1}^{\infty }i^{10}{\frac {x^{i}}{1-x^{i}}}\,$	$\zeta (s)\,\zeta (s-10)?\,$
11	$\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{11(\alpha _{i}+1)}-1}{{p_{i}}^{11}-1}}\,$	$\sum _{i=1}^{\infty }i^{11}{\frac {x^{i}}{1-x^{i}}}\,$	$\zeta (s)\,\zeta (s-11)?\,$
12	$\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{12(\alpha _{i}+1)}-1}{{p_{i}}^{12}-1}}\,$	$\sum _{i=1}^{\infty }i^{12}{\frac {x^{i}}{1-x^{i}}}\,$	$\zeta (s)\,\zeta (s-12)?\,$

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.