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# 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 .

Buchstab Function, Golomb-Dickman Constant, Greatest Prime Factor, Prime Factor

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## 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.

Dickman Function

## Cite this as:

Weisstein, Eric W. "Dickman Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DickmanFunction.html