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

(1)

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

			(2)
			(3)
			(4)

(Knuth 1998).

Amazingly, the average value of  such that  is

			(5)
			(6)
			(7)
			(8)
			(9)

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

(10)

Now change variables under the integral sign by defining

			(11)
			(12)

so

(13)

Plugging back in gives

(14)

To get rid of the s, define , so

(15)

But by the first fundamental theorem of calculus,

(16)

so differentiating both sides of equation (15) gives

(17)

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

(18)

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

(19)

for .

REFERENCES:

Dickman, K. "On the Frequency of Numbers Containing Prime Factors of a Certain Relative Magnitude." Arkiv för Mat., Astron. och Fys. 22A, 1-14, 1930.

Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, pp. 382-384, 1998.

Norton, K. K. Numbers with Small Prime Factors, and the Least kth Power Non-Residue. Providence, RI: Amer. Math. Soc., 1971.

Panario, D. "Smallest Components in Combinatorial Structures." Feb. 16, 1998. http://algo.inria.fr/seminars/sem97-98/panario.pdf.

Ramaswami, V. "On the Number of Positive Integers Less than  and Free of Prime Divisors Greater than ." Bull. Amer. Math. Soc. 55, 1122-1127, 1949.

Ramaswami, V. "The Number of Positive Integers  and Free of Prime Divisors , and a Problem of S. S. Pillai." Duke Math. J. 16, 99-109, 1949.