def test_primeomega():
assert primeomega(2) == 1
assert primeomega(2 * 2) == 2
assert primeomega(2 * 2 * 3) == 3
assert primeomega(3 * 25) == primeomega(3) + primeomega(25)
assert [primeomega(p) for p in primerange(1, 10)] == [1, 1, 1, 1]
assert primeomega(fac(50)) == 108
assert primeomega(2**9941 - 1) == 1
n = Symbol('n', integer=True)
assert primeomega(n)
assert primeomega(n).subs(n, 2**31 - 1) == 1
assert summation(primeomega(n), (n, 2, 30)) == 59
def test_factorint():
assert primefactors(123456) == [2, 3, 643]
assert factorint(0) == {0: 1}
assert factorint(1) == {}
assert factorint(-1) == {-1: 1}
assert factorint(-2) == {-1: 1, 2: 1}
assert factorint(-16) == {-1: 1, 2: 4}
assert factorint(2) == {2: 1}
assert factorint(126) == {2: 1, 3: 2, 7: 1}
assert factorint(123456) == {2: 6, 3: 1, 643: 1}
assert factorint(5951757) == {3: 1, 7: 1, 29: 2, 337: 1}
assert factorint(64015937) == {7993: 1, 8009: 1}
assert factorint(2**(2**6) + 1) == {274177: 1, 67280421310721: 1}
#issue 19683
assert factorint(10**38 - 1) == {
3: 2,
11: 1,
909090909090909091: 1,
1111111111111111111: 1
}
#issue 17676
assert factorint(28300421052393658575) == {
3: 1,
5: 2,
11: 2,
43: 1,
2063: 2,
4127: 1,
4129: 1
}
assert factorint(2063**2 * 4127**1 * 4129**1) == {
2063: 2,
4127: 1,
4129: 1
}
assert factorint(2347**2 * 7039**1 * 7043**1) == {
2347: 2,
7039: 1,
7043: 1
}
assert factorint(0, multiple=True) == [0]
assert factorint(1, multiple=True) == []
assert factorint(-1, multiple=True) == [-1]
assert factorint(-2, multiple=True) == [-1, 2]
assert factorint(-16, multiple=True) == [-1, 2, 2, 2, 2]
assert factorint(2, multiple=True) == [2]
assert factorint(24, multiple=True) == [2, 2, 2, 3]
assert factorint(126, multiple=True) == [2, 3, 3, 7]
assert factorint(123456, multiple=True) == [2, 2, 2, 2, 2, 2, 3, 643]
assert factorint(5951757, multiple=True) == [3, 7, 29, 29, 337]
assert factorint(64015937, multiple=True) == [7993, 8009]
assert factorint(2**(2**6) + 1, multiple=True) == [274177, 67280421310721]
assert factorint(fac(1, evaluate=False)) == {}
assert factorint(fac(7, evaluate=False)) == {2: 4, 3: 2, 5: 1, 7: 1}
assert factorint(fac(15, evaluate=False)) == \
{2: 11, 3: 6, 5: 3, 7: 2, 11: 1, 13: 1}
assert factorint(fac(20, evaluate=False)) == \
{2: 18, 3: 8, 5: 4, 7: 2, 11: 1, 13: 1, 17: 1, 19: 1}
assert factorint(fac(23, evaluate=False)) == \
{2: 19, 3: 9, 5: 4, 7: 3, 11: 2, 13: 1, 17: 1, 19: 1, 23: 1}
assert multiproduct(factorint(fac(200))) == fac(200)
assert multiproduct(factorint(fac(200, evaluate=False))) == fac(200)
for b, e in factorint(fac(150)).items():
assert e == fac_multiplicity(150, b)
for b, e in factorint(fac(150, evaluate=False)).items():
assert e == fac_multiplicity(150, b)
assert factorint(103005006059**7) == {103005006059: 7}
assert factorint(31337**191) == {31337: 191}
assert factorint(2**1000 * 3**500 * 257**127 * 383**60) == \
{2: 1000, 3: 500, 257: 127, 383: 60}
assert len(factorint(fac(10000))) == 1229
assert len(factorint(fac(10000, evaluate=False))) == 1229
assert factorint(12932983746293756928584532764589230) == \
{2: 1, 5: 1, 73: 1, 727719592270351: 1, 63564265087747: 1, 383: 1}
assert factorint(727719592270351) == {727719592270351: 1}
assert factorint(2**64 + 1, use_trial=False) == factorint(2**64 + 1)
for n in range(60000):
assert multiproduct(factorint(n)) == n
assert pollard_rho(2**64 + 1, seed=1) == 274177
assert pollard_rho(19, seed=1) is None
assert factorint(3, limit=2) == {3: 1}
assert factorint(12345) == {3: 1, 5: 1, 823: 1}
assert factorint(12345, limit=3) == {
4115: 1,
3: 1
} # the 5 is greater than the limit
assert factorint(1, limit=1) == {}
assert factorint(0, 3) == {0: 1}
assert factorint(12, limit=1) == {12: 1}
assert factorint(30, limit=2) == {2: 1, 15: 1}
assert factorint(16, limit=2) == {2: 4}
assert factorint(124, limit=3) == {2: 2, 31: 1}
assert factorint(4 * 31**2, limit=3) == {2: 2, 31: 2}
p1 = nextprime(2**32)
p2 = nextprime(2**16)
p3 = nextprime(p2)
assert factorint(p1 * p2 * p3) == {p1: 1, p2: 1, p3: 1}
assert factorint(13 * 17 * 19, limit=15) == {13: 1, 17 * 19: 1}
assert factorint(1951 * 15013 * 15053, limit=2000) == {
225990689: 1,
1951: 1
}
assert factorint(primorial(17) + 1, use_pm1=0) == \
{int(19026377261): 1, 3467: 1, 277: 1, 105229: 1}
# when prime b is closer than approx sqrt(8*p) to prime p then they are
# "close" and have a trivial factorization
a = nextprime(2**2**8) # 78 digits
b = nextprime(a + 2**2**4)
assert 'Fermat' in capture(lambda: factorint(a * b, verbose=1))
raises(ValueError, lambda: pollard_rho(4))
raises(ValueError, lambda: pollard_pm1(3))
raises(ValueError, lambda: pollard_pm1(10, B=2))
# verbose coverage
n = nextprime(2**16) * nextprime(2**17) * nextprime(1901)
assert 'with primes' in capture(lambda: factorint(n, verbose=1))
capture(lambda: factorint(nextprime(2**16) * 1012, verbose=1))
n = nextprime(2**17)
capture(lambda: factorint(n**3, verbose=1)) # perfect power termination
capture(lambda: factorint(2 * n, verbose=1)) # factoring complete msg
# exceed 1st
n = nextprime(2**17)
n *= nextprime(n)
assert '1000' in capture(lambda: factorint(n, limit=1000, verbose=1))
n *= nextprime(n)
assert len(factorint(n)) == 3
assert len(factorint(n, limit=p1)) == 3
n *= nextprime(2 * n)
# exceed 2nd
assert '2001' in capture(lambda: factorint(n, limit=2000, verbose=1))
assert capture(lambda: factorint(n, limit=4000, verbose=1)).count(
'Pollard') == 2
# non-prime pm1 result
n = nextprime(8069)
n *= nextprime(2 * n) * nextprime(2 * n, 2)
capture(lambda: factorint(n, verbose=1)) # non-prime pm1 result
# factor fermat composite
p1 = nextprime(2**17)
p2 = nextprime(2 * p1)
assert factorint((p1 * p2**2)**3) == {p1: 3, p2: 6}
# Test for non integer input
raises(ValueError, lambda: factorint(4.5))
# test dict/Dict input
sans = '2**10*3**3'
n = {4: 2, 12: 3}
assert str(factorint(n)) == sans
assert str(factorint(Dict(n))) == sans
def test_multiplicity_in_factorial():
n = fac(1000)
for i in (2, 4, 6, 12, 30, 36, 48, 60, 72, 96):
assert multiplicity(i, n) == multiplicity_in_factorial(i, 1000)