Sum of divisors function

This article needs more work.

Please help by expanding it!

The divisor function

σk (n)

for a positive integer

n

is defined as the sum of the

k

th powers,

k ∈ ℤ

, of the divisors of

n

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

For

k = 1

we get

$\sigma _{1}(n):=\sum _{d|n}d^{1}=\sum _{d|n}d=:\sigma (n),\,$

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

.
A000203

σ (n) =

sum of divisors of

n, n   ≥   1

. Also called

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, ...}

Formulae for the sum of divisors function

From the prime factorization of

n

$n=\prod _{i=1 \atop {{p_{i}}^{\alpha _{i}}\parallel n,\,\alpha _{i}\geq 1}}^{\omega (n)}{p_{i}}^{\alpha _{i}},\,$

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 sum of divisors of

n

$\sigma _{1}(n)={\frac {p_{i}^{\alpha _{i}+1}-1}{p_{i}-1}},\,$

since for each

pi

we can choose the exponent from 0 to

αi

to build a divisor of

n

.

Generating function of sum of divisors function

The generating function is

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

Dirichlet generating function of sum of divisors function

The Dirichlet generating function is

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

Sum of aliquot divisors of n

The aliquot divisors (or aliquot parts, and unfortunately referred to as proper divisors or proper parts) of

n

are the divisors of

n

less than

n

. The sum of aliquot divisors (sum of aliquot parts, and unfortunately referred to as sum of proper divisors or sum of proper parts) of

n

is then

$s(n)\equiv \sigma (n)-n.\,$

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. 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) of

n

are the divisors of

n

other than 1 or

n

. The sum of nontrivial divisors (sum of nontrivial parts, which should have been referred to as sum of proper divisors or sum of proper parts) of

n

is then

$s^{*}(n)\equiv s(n)-1=\sigma (n)-n-1.\,$

Perfect numbers

Perfect numbers are numbers such that

$\sigma (n)=2n,\,$

or equivalently

$s(n)=n.\,$

Multiperfect numbers

Category:Multiperfect numbers are numbers such that

$\sigma (n)=kn,\ k\geq 3.\,$

Deficient numbers

Deficient numbers are numbers such that

$\sigma (n)<2n,\,$

or equivalently

$s(n)<n.\,$

Abundant numbers

Abundant numbers are numbers such that

$\sigma (n)>2n,\,$

or equivalently

$s(n)>n.\,$

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

• Numbers with relatively many and large divisors

• Divisor function
  • Number of divisors function (number of divisors)
  • Sum of divisors function (sum of divisors)
  • Divisorial function (divisorial, product of divisors)

• Even divisors function
  • Number of even divisors function (number of even divisors)
  • Sum of even divisors function (sum of even divisors)
  • Product of even divisors function (product of even divisors)

• Odd divisors function
  • Number of odd divisors function (number of odd divisors)
    • Number of divisors of form 4 m + 1
    • Number of divisors of form 4 m + 3
    • (number of divisors of form 4 m + 1)  −  (number of divisors of form 4 m + 3)
  • Sum of odd divisors function (sum of odd divisors)
    • Sum of divisors of form 4 m + 1
    • Sum of divisors of form 4 m + 3
    • (sum of divisors of form 4 m + 1)  −  (sum of divisors of form 4 m + 3)
  • Product of odd divisors function (product of odd divisors)
    • Product of divisors of form 4 m + 1
    • Product of divisors of form 4 m + 3
    • (product of divisors of form 4 m + 1)  −  (product of divisors of form 4 m + 3)

• Sum of remainders function

Arithmetic function templates

• {{divisor function}} or {{sigma k}} arithmetic function template (for $\scriptstyle k\,\neq \,0\,$)
• {{number of divisors}} or {{sigma 0}} or {{tau}} arithmetic function template (for $\scriptstyle k\,=\,0\,$)
• {{sum of divisors}} or {{sigma 1}} or {{sigma}} arithmetic function template (for $\scriptstyle k\,=\,1\,$)