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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

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
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
since for each
pi
we can choose the exponent from 0 to
αi
to build a divisor of
n
, and which simplifies to

Generating function of the divisor function

The generating function is

Dirichlet generating function of the divisor function

The Dirichlet generating function is

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

For
k = −1
we get
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) = kn
.

σ0(n): Number of divisors function

For
k = 0
we get the number of divisors
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
)
where
σ0 (n)
is the number of divisors function.

σ1(n): Sum of divisors function

For
k = 1
we get the sum of divisors
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
)
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
pik  (αi +1) − 1
pik − 1
 , k ≥ 1.
Generating
function

G{σk (n)}(x) =


i  = 1
i  k
xi
1 − xi
Dirichlet
generating
function
D{σk (n)}(s) =


ζ (s) ζ (sk)?
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
1
2
3
4
5
6
7
8
9
10
11
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




Arithmetic function templates

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.