Help:=proc(): print(`ER(x,r), ERtest(x,r)`): 
print(`AllWalls(S,r)`):
end:
with(combinat):

#ER(x,r): the lower bound for the number of
#integers <=x that are not divisible by the
#first r primes:
ER:=proc(x,r) local gu,P,i,p:

P:=[ seq(ithprime(i),i=1..r)]:
x*mul(1-1/p,p in P)-2^r:

end:

#The exact number of integres <=x not
#divisible by the first r primes
ERtest:=proc(x,r) local i,j:
nops({seq(i,i=1..x)} minus 
{seq(seq(ithprime(i)*j,i=1..r),j=1..x)}):


end:


#AllWalls(S,r): the collection of all subsets
#of {1, ..., r} that contain S 
#AllWals({1,3},4) should be 
#{{1,3},{1,2,3},{1,3,4},{1,2,3,4}}
AllWalls:=proc(S,r) local michael,i,m:
michael:=powerset({seq(i,i=1..r)} minus S):
{seq(S union m, m in michael)}:

end:

#BrunGame1: inputs an integer x and  a set of
#sets-of-primes such that we know that
#NumberOfIntegers x not divisible by the
#first r prime >x*Sum((-1)^|S1|/mul(p,S1),S1 in S)-nops(S)
#and tries to remove free-loading walks with "apex"
#of even cardinality 
BrunGame1:=proc(x,r,S) local T,S1,dennis, d:

for S1 in S do
 if nops(S1) mod 2 =0 then
   dennis:=AllWalls(S1,r):
     for d in dennis do
       if FreeLoader(d,x,r) then
         RETURN(S minus d):
       fi:
   od:
 fi:
od:

#TO be continued
end: