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# Sum of divisors function

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
${\displaystyle \sigma _{k}(n):=\sum _{d|n}d^{k},\quad k\in \mathbb {Z} .\,}$
For
 k = 1
we get
${\displaystyle \sigma _{1}(n):=\sum _{d|n}d^{1}=\sum _{d|n}d=:\sigma (n),\,}$
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
${\displaystyle n=\prod _{i=1 \atop {{p_{i}}^{\alpha _{i}}\parallel n,\,\alpha _{i}\geq 1}}^{\omega (n)}{p_{i}}^{\alpha _{i}},\,}$
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
${\displaystyle \sigma _{1}(n)={\frac {p_{i}^{\alpha _{i}+1}-1}{p_{i}-1}},\,}$
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

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

### Dirichlet generating function of sum of divisors function

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

### 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
${\displaystyle s(n)\equiv \sigma (n)-n.\,}$

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

### Perfect numbers

Perfect numbers are numbers such that

${\displaystyle \sigma (n)=2n,\,}$

or equivalently

${\displaystyle s(n)=n.\,}$

### Multiperfect numbers

Category:Multiperfect numbers are numbers such that

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

### Deficient numbers

Deficient numbers are numbers such that

${\displaystyle \sigma (n)<2n,\,}$

or equivalently

${\displaystyle s(n)

### Abundant numbers

Abundant numbers are numbers such that

${\displaystyle \sigma (n)>2n,\,}$

or equivalently

${\displaystyle s(n)>n.\,}$

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## Sum of odd divisors

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### (sum of divisors of form 4m + 1) − (sum of divisors of form 4m + 3)

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