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
/t](https://mathworld.wolfram.com/images/equations/DickmanFunction/NumberedEquation9.svg) | (19) |
for 
.
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.