On the basis of rather convincing numerical evidence (iterative optimisation that always converges to the same place regardless of starting point), I conjecture that the worst case is
$$ a_i = \frac{i}{(n!)^{1/n}}.$$

I believe that gives
$$C(n) = (n-1) \biggl( \frac{2}{n(n+1)!} \biggr)^{1/(n-1)}
= e - \frac{e(7\ln n-\ln 2+\ln\pi)}{2n} + O((\ln n)^2/n^2).$$

The method is as boring as can be imagined. Start with a random vector **a** then make random changes to it, rejecting the change if the ratio of left side to right side gets greater. Stop when it hasn't changed for quite a while. Here is Maple code and sample output for $n=7$. Note that the vector is written with normalisation $a_1=1$ even though the function is calculated with $\prod a_i=1$; this is to make it easy to verify the conjecture by eye. Increase the value of *Digits* to get more precise verification.

```
# C(a) = left side divided by right side when a is normalised to product 1
C := proc(a::list) local b,s,k,n;
n := numelems(a);
s := mul(a)^(1/n);
add((1/k-2/n/(n+1))*a[k]/s,k=1..n) / add(k^2*s/a[k],k=1..n)^(1/(n-1));
evalf(%);
end proc:
eps := rand(-0.005..0.005):
r := rand(0.0..1.0):
n := 7; # Insert value of n here
p := rand(1..n):
a := [seq(r(),i=1..n)]:
Ca := C(a):
# Initial values
a,Ca;
while true do
currC := Ca:
for iter to 5000 do
a := a / a[1]; # Normalise to a[1]=1, note definition of C()
pos := p():
old := a[pos]:
a[pos] := max(old+eps(),0.0):
newCa := C(a):
if newCa < Ca then Ca := newCa: else a[pos] := old: end if:
end do:
print(a,Ca);
if Ca = currC then break end if:
end do:
# Compare to conjecture
evalf((n-1)*(2/(n*(n+1)!))^(1/(n-1)));
n := 7
[0.01506015221, 0.07126263187, 0.6247572599, 0.8230410086, 0.3693148049, 0.7607420702, 0.4647867183], 1.130896511
[1.000000000, 2.539500613, 16.40072680, 21.91302565, 10.73273281, 20.16793736, 13.31584408], 0.9921703822
[1.000000000, 1.724599348, 6.508732912, 8.844028091, 5.179715013, 8.535313648, 6.434865587], 0.8859967880
[1.000000000, 1.922974212, 3.388120139, 5.041522591, 4.236184244, 6.150065920, 4.972451482], 0.8403143690
[0.9999999998, 2.003100092, 3.006977468, 4.342457370, 5.023819919, 6.039027264, 6.158267516], 0.8324458625
[1.000000000, 2.000240008, 3.000252639, 4.000077928, 5.000611566, 6.000674528, 7.000510740], 0.8315464026
[1.000000000, 2.000204912, 3.000252639, 4.000077928, 5.000451793, 6.000674528, 7.000510740], 0.8315464022
[1.000000000, 2.000204912, 3.000252639, 4.000077928, 5.000451793, 6.000674528, 7.000510740], 0.8315464022
0.8315464026
```