Golomb–Dickman constant

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In mathematics, the Golomb–Dickman constant arises in the theory of random permutations and in number theory. Its value is

Definitions[edit]

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

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. More precisely,

where is the largest prime factor of k. So if k is a d digit integer, then 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 . More precisely,

where is the second largest prime factor n.

Formulae[edit]

There are several expressions for . Namely,

where is the exponential integral,

and

where is the Dickman function.

See also[edit]

External links[edit]