# Finding lower bounds of a combinatoric function.

I am working on a counting function involving prime numbers and managed to construct a sage function that is correct:

```
def sum_X_subs(plist, plen):
psum = 0
for k in range(plen-1):
ptwo = 2^(plen-k) - 2
psum += sum(mul(psub)*ptwo for psub in Subsets(plist, k))
return psum
# end sum_X_subs
flist = []
plist = [5]; plen = 1
for p in prime_range(7, 41+1):
plist.append(p)
plen += 1
flist.append(sum_X_subs(plist, plen))
flist
# [2, 52, 1162, 28196, 712250, 20379100, 696733730, 25016026280, 1042543611410, 47439135073960]
```

Now I need a lower bounds function for this: $$ S = prime~range(5, n)$$ $$f(S) = \sum_{k=0}^{|S|-1} {\left(\left(2^{|S|-k}-2\right)\sum_{T\in \mathcal{P}_{k}(S)} {\prod_{t\in T} {t}}\right)}$$ None of the literature I've found covers anything close to it. (I have a vague memory of an Euler project problem that involved subset products).