def sigma(n, x=1):
'''sigma(int) -> int
Divisor's sigma function'''
if n == 1:
return 1
return product(sumexp(p**x, k) for p, k in group(factorized(n)))
def divisors(n):
'''divisors(number) -> list
Positive divisors of the number, e.g. divisors(15) -> [1, 3, 5, 15]'''
if not n:
return []
factor_set = ({p**i for i in range(k+1)} for p, k in group(factorized(n)))
return sorted(product(c) for c in cartesian_product(*factor_set))
def phi(n):
'''Duality number-function data type.
as number: phi -> 1.618033988...
Golden Ratio, real number where phi == 1 + 1/phi.
as function: phi(number) -> number
Euler's totient, number of all 0 < a < n where gcd(a, n) == 1.'''
if n == 1:
return 1
return product((p-1) * p**(k-1) for p, k in group(factorized(n)))