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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
σ0(n)
is a count of the divisors of
n
and is sometimes notated
τ (n)
or
d (n)
instead.
σ1(n)
adds up the divisors of
n
and is often notated just
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

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,
n
is a k-perfect number since
σ1(n) = kn
.

Number of divisors function

For
k = 0
we get
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.

Sum of divisors function

For
k = 1
we get
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
.

Table of related formulae and values

Divisor function related formulae and values
Formulae

Generating

function

Dirichlet

generating

function

Differences

Partial sums

Partial sums of reciprocals

Sum of reciprocals

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.