divisor {elliptic} | R Documentation |

Various useful number theoretic functions

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

`n,k` |
Integers |

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.

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).

Robin K. S. Hankin and Bill Venables (`primes()`

and
`factorize()`

)

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

```
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?")
```

[Package *elliptic* version 1.4-0 Index]