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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) |
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(3) |
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(4) |
(Knuth 1998).
Amazingly, the average value of such that is
(5) |
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(6) |
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(7) |
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(8) |
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(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) |
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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.
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