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# Divisor function

(Redirected from Harmonic sum of divisors)

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

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
${\displaystyle n=\prod _{\stackrel {i=1}{{p_{i}}^{\alpha _{i}}\parallel \,n,\,\alpha _{i}\geq 1}}^{\omega (n)}{p_{i}}^{\alpha _{i}},\,}$
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
${\displaystyle \sigma _{k}(n)=\prod _{i=1}^{\omega (n)}\sum _{j=0}^{\alpha _{i}}{p_{i}}^{jk}=\prod _{i=1}^{\omega (n)}(1+{p_{i}}^{k}+{p_{i}}^{2k}+\cdots +{p_{i}}^{\alpha _{i}k})\,}$
since for each
 pi
we can choose the exponent from 0 to
 αi
to build a divisor of
 n
, and which simplifies to
${\displaystyle \sigma _{k}(n)={\begin{cases}\prod _{i=1}^{\omega (n)}(1+\alpha _{i})&{\text{if }}k=0,\\\prod _{i=1}^{\omega (n)}{\frac {{p_{i}}^{(\alpha _{i}+1)k}-1}{{p_{i}}^{k}-1}}&{\text{if }}k>0.\end{cases}}}$

## Generating function of the divisor function

The generating function is

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

## Dirichlet generating function of the divisor function

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

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

For
 k = −1
we get
${\displaystyle \sigma _{-1}(n):=\sum _{d|n}d^{-1}=\sum _{d|n}{\frac {1}{d}}=\sum _{d|n}{\frac {1}{n/d}}=\sum _{d|n}{\frac {d}{n}}={\frac {1}{n}}\sum _{d|n}d={\frac {\sigma _{1}(n)}{n}},\,}$
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
${\displaystyle \sigma _{0}(n):=\sum _{d|n}d^{0}=\sum _{d|n}1=:D(n)=:d(n)=:\nu (n)=:\tau (n),\,}$
where
 τ (n)
is the number of divisors function. The notations
 d (n)
[1],
 ν (n)
[2], and
 τ (n)
[3] 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
)
${\displaystyle s_{0}(n):=\sigma _{0}(n)-1,\,}$
where
 σ0 (n)
is the number of divisors function.

## σ1(n): Sum of divisors function

For
 k = 1
we get the sum of divisors
${\displaystyle \sigma _{1}(n):=\sum _{d|n}d^{1}=\sum _{d|n}d=:\sigma (n),\,}$
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
)
${\displaystyle s_{1}(n):=\sigma _{1}(n)-n,\,}$
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 ${\displaystyle \prod _{i=1}^{\omega (n)}(\alpha _{i}+1)\,}$ ${\displaystyle \sum _{i=1}^{\infty }{\frac {x^{i}}{1-x^{i}}}\,}$ ${\displaystyle (\zeta (s))^{2}\,}$
1 ${\displaystyle \prod _{i=1}^{\omega (n)}{\frac {p_{i}^{\alpha _{i}+1}-1}{p_{i}-1}}\,}$ ${\displaystyle \sum _{i=1}^{\infty }i{\frac {x^{i}}{1-x^{i}}}\,}$ ${\displaystyle \zeta (s)\,\zeta (s-1)\,}$
2 ${\displaystyle \prod _{i=1}^{\omega (n)}{\frac {p_{i}^{2(\alpha _{i}+1)}-1}{{p_{i}}^{2}-1}}\,}$ ${\displaystyle \sum _{i=1}^{\infty }i^{2}{\frac {x^{i}}{1-x^{i}}}\,}$ ${\displaystyle \zeta (s)\,\zeta (s-2)\,}$
3 ${\displaystyle \prod _{i=1}^{\omega (n)}{\frac {p_{i}^{3(\alpha _{i}+1)}-1}{{p_{i}}^{3}-1}}\,}$ ${\displaystyle \sum _{i=1}^{\infty }i^{3}{\frac {x^{i}}{1-x^{i}}}\,}$ ${\displaystyle \zeta (s)\,\zeta (s-3)?\,}$
4 ${\displaystyle \prod _{i=1}^{\omega (n)}{\frac {p_{i}^{4(\alpha _{i}+1)}-1}{{p_{i}}^{4}-1}}\,}$ ${\displaystyle \sum _{i=1}^{\infty }i^{4}{\frac {x^{i}}{1-x^{i}}}\,}$ ${\displaystyle \zeta (s)\,\zeta (s-4)?\,}$
5 ${\displaystyle \prod _{i=1}^{\omega (n)}{\frac {p_{i}^{5(\alpha _{i}+1)}-1}{{p_{i}}^{5}-1}}\,}$ ${\displaystyle \sum _{i=1}^{\infty }i^{5}{\frac {x^{i}}{1-x^{i}}}\,}$ ${\displaystyle \zeta (s)\,\zeta (s-5)?\,}$
6 ${\displaystyle \prod _{i=1}^{\omega (n)}{\frac {p_{i}^{6(\alpha _{i}+1)}-1}{{p_{i}}^{6}-1}}\,}$ ${\displaystyle \sum _{i=1}^{\infty }i^{6}{\frac {x^{i}}{1-x^{i}}}\,}$ ${\displaystyle \zeta (s)\,\zeta (s-6)?\,}$
7 ${\displaystyle \prod _{i=1}^{\omega (n)}{\frac {p_{i}^{7(\alpha _{i}+1)}-1}{{p_{i}}^{7}-1}}\,}$ ${\displaystyle \sum _{i=1}^{\infty }i^{7}{\frac {x^{i}}{1-x^{i}}}\,}$ ${\displaystyle \zeta (s)\,\zeta (s-7)?\,}$
8 ${\displaystyle \prod _{i=1}^{\omega (n)}{\frac {p_{i}^{8(\alpha _{i}+1)}-1}{{p_{i}}^{8}-1}}\,}$ ${\displaystyle \sum _{i=1}^{\infty }i^{8}{\frac {x^{i}}{1-x^{i}}}\,}$ ${\displaystyle \zeta (s)\,\zeta (s-8)?\,}$
9 ${\displaystyle \prod _{i=1}^{\omega (n)}{\frac {p_{i}^{9(\alpha _{i}+1)}-1}{{p_{i}}^{9}-1}}\,}$ ${\displaystyle \sum _{i=1}^{\infty }i^{9}{\frac {x^{i}}{1-x^{i}}}\,}$ ${\displaystyle \zeta (s)\,\zeta (s-9)?\,}$
10 ${\displaystyle \prod _{i=1}^{\omega (n)}{\frac {p_{i}^{10(\alpha _{i}+1)}-1}{{p_{i}}^{10}-1}}\,}$ ${\displaystyle \sum _{i=1}^{\infty }i^{10}{\frac {x^{i}}{1-x^{i}}}\,}$ ${\displaystyle \zeta (s)\,\zeta (s-10)?\,}$
11 ${\displaystyle \prod _{i=1}^{\omega (n)}{\frac {p_{i}^{11(\alpha _{i}+1)}-1}{{p_{i}}^{11}-1}}\,}$ ${\displaystyle \sum _{i=1}^{\infty }i^{11}{\frac {x^{i}}{1-x^{i}}}\,}$ ${\displaystyle \zeta (s)\,\zeta (s-11)?\,}$
12 ${\displaystyle \prod _{i=1}^{\omega (n)}{\frac {p_{i}^{12(\alpha _{i}+1)}-1}{{p_{i}}^{12}-1}}\,}$ ${\displaystyle \sum _{i=1}^{\infty }i^{12}{\frac {x^{i}}{1-x^{i}}}\,}$ ${\displaystyle \zeta (s)\,\zeta (s-12)?\,}$

## 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)

## Notes

1. Hardy and Wright 1979, p. 239.
2. Ore 1988, p. 86.
3. Burton 1989, p. 128.

## 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.