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Sum of divisors function
From OeisWiki
σk (n) |
n |
k |
k ∈ ℤ |
n |
k = 1 |
σ (n) |
σ (n) |
σ1 (n) |
n |
A000203
σ (n) = |
n, n ≥ 1 |
σ1 (n) |
- {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, ...}
Contents
- 1 Formulae for the sum of divisors function
- 2 Generating function of sum of divisors function
- 3 Dirichlet generating function of sum of divisors function
- 4 Sum of aliquot divisors of n
- 5 Sum of nontrivial divisors of n
- 6 Perfect numbers
- 7 Multiperfect numbers
- 8 Deficient numbers
- 9 Abundant numbers
- 10 Sum of even divisors
- 11 Sum of odd divisors
- 12 See also
- 13 Notes
Formulae for the sum of divisors function
From the prime factorization ofn |
pi |
n |
ω (n) |
n |
n |
pi |
αi |
n |
Generating function of sum of divisors function
The generating function is
Dirichlet generating function of sum of divisors function
The Dirichlet generating function is
Sum of aliquot divisors of n
The aliquot divisors (or aliquot parts, and unfortunately referred to as proper divisors or proper parts) ofn |
n |
n |
n |
Untouchable numbers
Some numbers never come up as aliquot sums, such as 5. If a stronger version of the Goldbach conjecture that says all even numbers greater than 6 are the sum of two primes is proven true, that would also prove that 5 is the only odd untouchable number.^{[1]} See A005114.
Sum of nontrivial divisors of n
The nontrivial divisors (or nontrivial parts, which should have been referred to as proper divisors or proper parts) ofn |
n |
n |
n |
Perfect numbers
Perfect numbers are numbers such that
or equivalently
Multiperfect numbers
Category:Multiperfect numbers are numbers such that
Deficient numbers
Deficient numbers are numbers such that
or equivalently
Abundant numbers
Abundant numbers are numbers such that
or equivalently
Sum of even divisors
(...)
Sum of odd divisors
(...)
Sum of divisors of form 4m + 1
(...)
Sum of divisors of form 4m + 3
(...)
(sum of divisors of form 4m + 1) − (sum of divisors of form 4m + 3)
(...)
See also
- Odd divisors function
Arithmetic function templates
- {{divisor function}} or {{sigma k}} arithmetic function template (for )
- {{number of divisors}} or {{sigma 0}} or {{tau}} arithmetic function template (for )
- {{sum of divisors}} or {{sigma 1}} or {{sigma}} arithmetic function template (for )
Notes
- ↑ Adams-Watters, Frank and Weisstein, Eric W. "Untouchable Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/UntouchableNumber.html