#Code done in Dr. Z.'s Experimental Math Class, Spring 2016, Jan. 21, 206
Help:=proc() : print(` IsPerfect(n) , PerfectL(N) ,coin() `) : 
print(` Game(N) , Mer(N), Hn(n)  , JL(n)`):
end:

with(numtheory):

IsPerfect:=proc(n)  evalb(sigma(n)=2*n): end:

#PerfectL(N): the list of all the perfect numbers <=N
PerfectL:=proc(N) local n,L:

L:=[]:

for n from 1 to N do

   if IsPerfect(n) then
     L:=[op(L), n]:
   fi:

od:

L:
end:

#coin(): tossing a coin, outputs 1 or -1 with prob. 1/2
coin:=proc() : 2*rand(0..1)()-1: end:

#Game(N): you toss a fair coin N times and see how much you won
#or lost
Game:=proc(N) local i:
add(coin(),i=1..N):
end:

#Mer(N):Mertens(N): the sum of mobius(i) from i=1 to i=N
Mer:=proc(N) local i: add(mobius(i), i=1..N): end:

#Hn(n): the Harmonic numbers
Hn:=proc(n) local i: add(1/i,i=1..n): end:

#JL(n):
JL:=proc(n)

evalf(Hn(n)+exp(Hn(n))*log(Hn(n))-sigma(n),20):

end: