# Dickman-de Bruijn function

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## Definition

This function, called Dickman's function or the Dickman-de Bruijn function, is defined as the function  satisfying the delay differential equation:



subject to the initial condition  for . The function satisfies the following properties:

•  for .
•  for .
•  is (strictly) decreasing for , i.e.,  for .
•  is once differentiable on . More generally, \$\rho[/itex] is  times differentiable everywhere except at the points .
•  is infinitely differentiable except at integers.
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## Related facts

It turns out that the density of numbers with no prime divisor greater than the  root is given by . Formally, consider, for any , the fraction of natural numbers  such that all prime divisors of  are at most . Then, as , this fraction tends to .