# divisor

##### Number theoretic functions

Various useful number theoretic functions

- Keywords
- math

##### Usage

```
divisor(n,k=1)
primes(n)
factorize(n)
mobius(n)
totient(n)
liouville(n)
```

##### Arguments

- n,k
Integers

##### Details

Functions `primes()`

and `factorize()`

cut-and-pasted from
Bill Venables's conf.design package, version 0.0-3. Function
`primes(n)`

returns a vector of all primes not exceeding
`n`

; function `factorize(n)`

returns an integer vector of
nondecreasing primes whose product is `n`

.

The others are multiplicative functions, defined in Hardy and Wright:

Function `divisor()`

, also written
\(\sigma_k(n)\), is the divisor function defined on
p239. This gives the sum of the \(k^{\rm th}\) powers of all
the divisors of `n`

. Setting \(k=0\) corresponds to
\(d(n)\), which gives the number of divisors of `n`

.

Function `mobius()`

is the Moebius function (p234), giving zero
if `n`

has a repeated prime factor, and \((-1)^q\) where
\(n=p_1p_2\ldots p_q\) otherwise.

Function `totient()`

is Euler's totient function (p52), giving
the number of integers smaller than `n`

and relatively prime to
it.

Function `liouville()`

gives the Liouville function.

##### Note

The divisor function crops up in `g2.fun()`

and `g3.fun()`

.
Note that this function is not called `sigma()`

to
avoid conflicts with Weierstrass's \(\sigma\) function (which
ought to take priority in this context).

##### References

G. H. Hardy and E. M. Wright, 1985. *An
introduction to the theory of numbers* (fifth edition).
Oxford University Press.

##### Examples

```
# NOT RUN {
mobius(1)
mobius(2)
divisor(140)
divisor(140,3)
plot(divisor(1:100,k=1),type="s",xlab="n",ylab="divisor(n,1)")
plot(cumsum(liouville(1:1000)),type="l",main="does the function ever exceed zero?")
# }
```

*Documentation reproduced from package elliptic, version 1.4-0, License: GPL-2*