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from hypothesis import given
from hypothesis import strategies as st
from sympy import divisors
from sympy.functions.combinatorial.numbers import divisor_sigma, totient
from sympy.ntheory.primetest import is_square
@given(n=st.integers(1, 10**10))
def test_tau_hypothesis(n):
div = divisors(n)
tau_n = len(div)
assert is_square(n) == (tau_n % 2 == 1)
sigmas = [divisor_sigma(i) for i in div]
totients = [totient(n // i) for i in div]
mul = [a * b for a, b in zip(sigmas, totients)]
assert n * tau_n == sum(mul)
@given(n=st.integers(1, 10**10))
def test_totient_hypothesis(n):
assert totient(n) <= n
div = divisors(n)
totients = [totient(i) for i in div]
assert n == sum(totients)
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