Let
an be the average — taken over all
permutations of a set of size
n — of the length of the longest
cycle in each permutation. Then the Golomb–Dickman constant is : \lambda = \lim_{n\to\infty} \frac{a_n}{n}. In the language of
probability theory, \lambda n 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. More precisely, :\lambda = \lim_{n\to\infty} \frac1n \sum_{k=2}^n \frac{\log(P_1(k))}{\log(k)}, where P_1(k) is the largest prime factor of
k . So if
k is a
d digit integer, then \lambda d 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? Asymptotically, this probability is \lambda. More precisely, :\lambda = \lim_{n\to\infty} \text{Prob}\left\{P_2(n) \le \sqrt{P_1(n)}\right\} where P_2(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 f^{n+k}(x) = f^n(x) 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 proved that : \lim_{n\to\infty} \frac{b_n}{\sqrt{n}} = \sqrt{\frac{\pi}{2} } \lambda. == Formulae ==