Dickman Function

The probability that a random integer between 1 and will have its greatest prime factor approaches a limiting value as , where for and is defined through the integral equation

for (Dickman 1930, Knuth 1998), which is almost (but not quite) a homogeneous Volterra integral equation of the second kind. The function can be given analytically for by

(Knuth 1998).

Amazingly, the average value of such that is

which is precisely the Golomb-Dickman constant , which is defined in a completely different way!

The Dickman function can be solved numerically by converting it to a delay differential equation. This can be done by noting that will become upon multiplicative inversion, so define to obtain

Now change variables under the integral sign by defining

so

Plugging back in gives

To get rid of the s, define , so

But by the first fundamental theorem of calculus,

so differentiating both sides of equation (15) gives

This holds for , which corresponds to . Rearranging and combining with an appropriate statement of the condition for in the new variables then gives

The second-largest prime factor will be is given by an expression similar to that for . It is denoted , where for and

for .