Help22:=proc(): print(`BAc(f), Likelyc(q,r), LikelycC(q,r)  `):end:

read `QC.txt`:

#BAc(f): BA(f) done cleverly. The best fraction approximation to the fraction f with denominator<=sqrt(denom(f))
BAc:=proc(f) local L,i:
L:=CVG(f):
for i from 1 to nops(L) while denom(L[i])^2<=denom(f) do od:
L[i-1]:
end:


#Likelyc(q,r): All the c less than q such that |mods(r*c,q)}<=r/2
Likelyc:=proc(q,r) local L,c:
L:=[]:

for c from 1 to q-1 do
 if abs(mods(r*c,q))<=r/2 then

  if c mod 2=0 then
   c:=c+1:
  fi:

   L:=[op(L),c]:
 fi:
od:
L:
end:

#LikelycC(q,r): All the closest integers to integer multiples of c.
LikelycC:=proc(q,r) local i,L,c:
L:=[]:
for i from 1 to r-1 do

c:=round((q/r)*i):
  if c mod 2=0 then
   c:=c+1:
  fi:

L:=[op(L),c]:
od:
L:
end: