# divisor

0th

Percentile

##### 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)
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.

• divisor
• primes
• factorize
• mobius
• totient
• liouville
##### 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

### Community examples

Looks like there are no examples yet.