# least prime factor

The least prime factor of a positive integer $n$ is the smallest positive prime number dividing $n$. Sometimes expressed as a function, $\textrm{lpf}(n)$. For example, $\textrm{lpf}(91)=7$. For a prime number $p$, clearly $\textrm{lpf}(p)=p$, while for any composite number (except squares of primes) $(\textrm{lpf}(n))^{2}. (The function would be quite useless if 1 is considered a prime, therefore $\textrm{lpf}(1)$ is undefined — though we could make an argument for $\textrm{lpf}(0)=2$). In the sequence of least prime factors for each integer in turn, each prime occurs first at the index for itself then not again until its square.

In Mathematica, one can use LeastPrimeFactor[n] after loading a number theory package, or much more simply by using the command FactorInteger[n][[1,1]] (of course substituting n as necessary).

Title least prime factor LeastPrimeFactor 2013-03-22 17:40:03 2013-03-22 17:40:03 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Definition msc 11A51