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Divisor function
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(Redirected from Sum of proper divisors)
σk (n), k ∈ ℤ, |
n |
k |
n |
In particular
-
is the harmonic sum of divisors ofσ − 1(n)
;n
-
is the number of divisors ofσ0(n)
, and is often notatedn
orτ (n)
;d (n) -
is the sum of divisors ofσ1(n)
and is often notatedn
.σ (n)
Contents
- 1 Formulae for the divisor function
- 2 Generating function of the divisor function
- 3 Dirichlet generating function of the divisor function
- 4 σ−1(n): Harmonic sum of divisors function
- 5 σ0(n): Number of divisors function
- 6 σ1(n): Sum of divisors function
- 7 Table of related formulae and values
- 8 Table of sequences
- 9 See also
- 10 Notes
- 11 References
Formulae for the divisor function
From the prime factorization ofn |
pi |
n |
ω (n) |
n |
n |
pi |
αi |
n |
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
Fork = −1 |
σ−1(n) |
n |
σ−1(n) = k |
k |
n |
σ1(n) = k n |
σ0(n): Number of divisors function
Fork = 0 |
τ (n) |
d (n) |
ν (n) |
τ (n) |
σ0(n) |
n |
n > 0 |
s0(n): Number of aliquot divisors function
Fork = 0 |
n |
σ0 (n) |
σ1(n): Sum of divisors function
Fork = 1 |
σ (n) |
σ (n) |
σ1(n) |
n |
s1(n): Sum of aliquot divisors function
Fork = 1 |
n |
σ1 (n) |
|
Formulae
|
Generating function
|
Dirichlet generating function
|
Differences
|
Partial sums
|
Partial sums of reciprocals
|
Sum of reciprocals
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1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
3 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
5 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
9 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
10 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
11 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
12 |
Table of sequences
|
|
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
| ||
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
| ||
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
| ||
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
| ||
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
| ||
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
| ||
6 | {1, 65, 730, 4161, 15626, 47450, 117650, 266305, 532171, 1015690, 1771562, 3037530, 4826810, 7647250, 11406980, 17043521, 24137570, 34591115, 47045882, 65019786, 85884500, ...}
|
A013954
| ||
7 | {1, 129, 2188, 16513, 78126, 282252, 823544, 2113665, 4785157, 10078254, 19487172, 36130444, 62748518, 106237176, 170939688, 270549121, 410338674, 617285253, 893871740, ...}
|
A013955
| ||
8 | {1, 257, 6562, 65793, 390626, 1686434, 5764802, 16843009, 43053283, 100390882, 214358882, 431733666, 815730722, 1481554114, 2563287812, 4311810305, 6975757442, ...}
|
A013956
| ||
9 | {1, 513, 19684, 262657, 1953126, 10097892, 40353608, 134480385, 387440173, 1001953638, 2357947692, 5170140388, 10604499374, 20701400904, 38445332184, 68853957121, ...}
|
A??????
| ||
10 | {1, 1025, ...}
|
A??????
| ||
11 | {1, 2049, ...}
|
A??????
| ||
12 | {1, 4097, ...}
|
A??????
|
See also
- Odd divisors function
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
Notes
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.