Golomb–Dickman constant

In mathematics, the Golomb–Dickman constant, named after Solomon W. Golomb and Karl Dickman, is a mathematical constant, which arises in the theory of random permutations and in number theory.

Then the Golomb–Dickman constant is In the language of probability theory,

is asymptotically the expected length of the longest cycle in a uniformly distributed random permutation of a set of size n. In number theory, the Golomb–Dickman constant appears in connection with the average size of the largest prime factor of an integer.

is the largest prime factor of k (sequence A006530 in the OEIS) .

is the asymptotic average number of digits of the largest prime factor of k. The Golomb–Dickman constant appears in number theory in a different way.

What is the probability that second largest prime factor of n is smaller than the square root of the largest prime factor of n?

is the second largest prime factor n. The Golomb-Dickman constant also arises when we consider the average length of the largest cycle of any function from a finite set to itself.

If X is a finite set, if we repeatedly apply a function f: X → X to any element x of this set, it eventually enters a cycle, meaning that for some k we have

for sufficiently large n; the smallest k with this property is the length of the cycle.

Let bn be the average, taken over all functions from a set of size n to itself, of the length of the largest cycle.

Then Purdom and Williams[3] proved that There are several expressions for