def PE(N): # @DuplicatedSignature
primes = primeSieve(N + 1) # returns list of all primes <=N
sqrtN = sqrt(N)
ans = 1
for p in primes:
if p <= sqrtN:
ans *= p ** (int(log(N) / log(p)))
else:
ans *= p
return ans
def lcm(N):
primes = primeSieve(N + 1)
sqrtN = sqrt(N)
ans = 1
for p in primes:
if p <= sqrtN:
ans *= p**(int(log(N) / log(p)))
else:
ans *= p
return ans
def primeFactors(n): # returns list of all prime factors
primes = primeSieve(int(math.ceil(math.sqrt(n))) + 1)
factors = list()
i = 0
fact = n
while fact not in primes and i < len(primes):
if fact % primes[i] == 0:
fact = fact / primes[i]
factors.append(primes[i])
else:
i += 1
factors.append(fact) # remaining factor must be prime so add
uniques = set(factors)
primeFactorisation = dict()
for prime in uniques:
exp = factors.count(prime)
primeFactorisation[prime] = exp
return primeFactorisation
