\name{sigma}
\alias{sigma}
\alias{tau}
\title{Divisor Functions}
\description{
  Sum of powers of all divisors of a natural number.
}
\usage{
sigma(n, k = 1, proper = FALSE)

tau(n)
}
\arguments{
  \item{n}{Positive integer.}
  \item{k}{Numeric scalar, the exponent to be used.}
  \item{proper}{Logical; if \code{TRUE}, n will \emph{not} be considered
                as a divisor of itself; default: FALSE.}
}
\details{
  Total sum of all integer divisors of \code{n} to the power of \code{k},
  including \code{1} and \code{n}.

  For \code{k=0} this is the number of divisors, for \code{k=1}
  it is the sum of all divisors of \code{n}.

  \code{tau} is Ramanujan`s \emph{tau} function, here computed using
  \code{sigma(., 5)} and \code{sigma(., 11)}.

  A number is called \emph{refactorable}, if \code{tau(n)} divides \code{n},
  for example \code{n=12} or \code{n=18}.
}
\value{
  Natural number, the number or sum of all divisors.
}
\author{
  HwB <hwborchers@googlemail.com>
}
\references{
\code{http://en.wikipedia.org/wiki/Divisor_function}

\code{http://en.wikipedia.org/wiki/Tau-function}
}
\note{
  Works well up to \code{10^9}.
}
\seealso{
\code{\link{ifactor}}
}
\examples{
sapply(1:16, sigma, k = 0)
sapply(1:16, sigma, k = 1)
sapply(1:16, sigma, proper = TRUE)
}
\keyword{ math }