# sigma: Divisor Functions In numbers: Number-Theoretic Functions

## Description

Sum of powers of all divisors of a natural number.

## Usage

 ```1 2 3``` ```Sigma(n, k = 1, proper = FALSE) tau(n) ```

## Arguments

 `n` Positive integer. `k` Numeric scalar, the exponent to be used. `proper` Logical; if `TRUE`, n will not be considered as a divisor of itself; default: FALSE.

## Details

Total sum of all integer divisors of `n` to the power of `k`, including `1` and `n`.

For `k=0` this is the number of divisors, for `k=1` it is the sum of all divisors of `n`.

`tau` is Ramanujan's tau function, here computed using `Sigma(., 5)` and `Sigma(., 11)`.

A number is called refactorable, if `tau(n)` divides `n`, for example `n=12` or `n=18`.

## Value

Natural number, the number or sum of all divisors.

## Note

Works well up to `10^9`.

## References

`https://en.wikipedia.org/wiki/Divisor_function`

`https://en.wikipedia.org/wiki/Ramanujan_tau_function`

`primeFactors`, `divisors`

## Examples

 ```1 2 3``` ```sapply(1:16, Sigma, k = 0) sapply(1:16, Sigma, k = 1) sapply(1:16, Sigma, proper = TRUE) ```

### Example output

``` [1] 1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5
[1]  1  3  4  7  6 12  8 15 13 18 12 28 14 24 24 31
[1]  0  1  1  3  1  6  1  7  4  8  1 16  1 10  9 15
```

numbers documentation built on May 15, 2021, 1:08 a.m.