Finding $k$-th term of the sequence. I am finding a specific term of the sequence of integers given three primes $p_1$, $p_2$, and $p_3$. The sequences consist of all possible products of $p_1$, $p_2$, and $p_3$ as many times including $0$, and $k$-th term is the $k$-th smallest product of those primes.
For example, if $p_1 = 2$, $p_2 = 3$, and $p_3 = 5$, the sequence looks like below.
$1,2,3,4,5,6,8,9,10,12 ...$
$7$-th term of the sequence is the $7$-th smallest element you can make by multiplying $2$, $3$ and $5$. That is $8$ as $8 = 2^3$. $8$-th term is given by $9 = 3^3$ and $9$-th term is given by $10 = 2^1 5^1$.
I want to find recurrence relation but can't find any.
 A: I don't think you can get a recurrence,
but you can get an estimate
for the number of $a_i \le n$
as a function of $n$.
Suppose there are
$k$ primes
$(p_i)_{i=1}^k
$.
Then you want
the number of the number of $a_j$
such that
$a_j
=\prod_{i=1}^k p_i^{r_i}
\le n$
where $r_i \ge 0$ are integers.
This is
$\sum_{i=1}^k r_i\ln(p_i)
\le \ln(n)$
and the number of these
lattice points is
(here's the big jump)
approximately
the volume of the 
k-dimensional simplex
$\sum_{i=1}^k x_i\ln(p_i)
\le \ln(n)$
where the $x_i$
are the $k$ ordinates.
The volume of this is
$\dfrac{\ln^k(n)}{k!\prod_{i=1}^k\ln(p_i)}
$
and this is the estimate
for the number of
$a_i \le n$.
This is a generalization
of a theorem of Ramanujan
(isn't everything?)
for the case
$k=2, p_1=2, p_2 = 3$
discussed by Hardy
starting on page 69
of his book on
12 lectures about
Ramanujan's work.
This is still available
and I have read it more then once.
https://www.amazon.com/Ramanujan-Lectures-Subjects-Suggested-Publishing/dp/0821820230/ref=sr_1_1?crid=OTBX1W9N70ET&dchild=1&keywords=ramanujan+hardy&qid=1587701839&sprefix=hardy+ram%2Caps%2C208&sr=8-1
