and 
be positive integers such that:
is divisible by 
is not divisible by 
.
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
,
![$P \in \mathbb Q[x]$](http://latex.artofproblemsolving.com/a/f/0/af08a4976a47f328ce0af014fc87fa559919bc24.png)
,
is super irreductible if ![$\forall Q \in \mathbb [x], \deg(Q)>0, P(Q(x))$](http://latex.artofproblemsolving.com/7/3/3/733796418869db0b29b473ce49d58b62d14cc6f7.png)
is irreductible.
Y by
Let 
be a positive integer, and let its positive divisors be
![\[
d_1 < d_2 < \cdots < d_k.
\]](//latex.artofproblemsolving.com/6/3/e/63ec82aba0d6c4eece89af6a1247a89498616281.png)
Define 
to be the number of ordered pairs 
with 
such that 
.
Find
.
Also, find a general formula for when
![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](//latex.artofproblemsolving.com/2/7/2/272dc2cc33ebbf695311cb61567fa0297e1299b6.png)
where the 
are distinct primes and the 
are positive integers.
be a positive integer, and let its positive divisors be![\[
d_1 < d_2 < \cdots < d_k.
\]](http://latex.artofproblemsolving.com/6/3/e/63ec82aba0d6c4eece89af6a1247a89498616281.png)
Define 
to be the number of ordered pairs 
with 
such that 
.Find
.Also, find a general formula for
when![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](http://latex.artofproblemsolving.com/2/7/2/272dc2cc33ebbf695311cb61567fa0297e1299b6.png)
where the 
are distinct primes and the 
are positive integers.
and all 
.
pairs 
with 
.
to 
where 
.
.
.
hence 
.
with the 
distinct. Then we count disjoint subsets 
of 
.
w
hence we find 
,
is multiplicative and 
.
