Help:=proc(): print(`Gen(S,c) ,IV(k,n)`): end:

#The multiplicative semi-group of integers modulo c
#generated by the integers in S
#For example Gen({2},10); should give {1,2,4,8,6}
Gen:=proc(S,c) local S1,S2,s,S3,s2:

S1:={1}:

S2:=S1 union S mod c:

while S1 <>S2 do

S3:={seq(seq(s*s2 mod c,s2 in S2),s in S)} union S2:

S1:=S2:

S2:=S3:

od:

S1: 

end:



#IV(k,n): the set of vectors of non-negative
#integers with k components that add-up to n
#For example, IV(1,3) should give {[3]}.
IV:=proc(k,n) local S,i,S1:
option remember:

if k<0 or n<0 then
  RETURN({}):
fi:

if k=0 then
 if n=0 then
   RETURN({[]}):
 else
   RETURN({}):
 fi:
fi:

S:={}:

#i =last component of the vector
 for i from 0 to n  do
  S1:=IV(k-1,n-i):
  S:=S union {seq([op(t),i],t in S1)}:
 od:

S:

end: