text: from sympy import Sieve, sieve, Symbol, S, limit, I, zoo, nan
  from sympy.core.compatibility import range
  
  from sympy.ntheory import isprime, totient, mobius, randprime, nextprime, prevprime, \
      primerange, primepi, prime, primorial, composite, compositepi, reduced_totient
  from sympy.ntheory.generate import cycle_length
  from sympy.ntheory.primetest import mr
  from sympy.utilities.pytest import raises
  
  from array import array as _array
  
  def test_prime():
      assert prime(1) == 2
      assert prime(2) == 3
      assert prime(5) == 11
      assert prime(11) == 31
      assert prime(57) == 269
      assert prime(296) == 1949
      assert prime(559) == 4051
      assert prime(3000) == 27449
      assert prime(4096) == 38873
      assert prime(9096) == 94321
      assert prime(25023) == 287341
      raises(ValueError, lambda: prime(0))
      sieve.extend(3000)
      assert prime(401) == 2749
  
  
  def test_primepi():
      assert primepi(-1) == 0
      assert primepi(1) == 0
      assert primepi(2) == 1
      assert primepi(S(7)/2) == 2
      assert primepi(3.5) == 2
      assert primepi(5) == 3
      assert primepi(11) == 5
      assert primepi(57) == 16
      assert primepi(296) == 62
      assert primepi(559) == 102
      assert primepi(3000) == 430
      assert primepi(4096) == 564
      assert primepi(9096) == 1128
      assert primepi(25023) == 2763
      assert primepi(10**8) == 5761455
      assert primepi(253425253) == 13856396
      assert primepi(8769575643) == 401464322
      sieve.extend(3000)
      assert primepi(2000) == 303
  
      n = Symbol('n')
      assert primepi(n).subs(n, 2) == 1
  
      r = Symbol('r', real=True)
      assert primepi(r).subs(r, 2) == 1
  
      assert primepi(S.Infinity) == S.Infinity
      assert primepi(-S.Infinity) == 0
  
      assert limit(primepi(n), n, 100) == 25
  
      raises(ValueError, lambda: primepi(I))
      raises(ValueError, lambda: primepi(1 + I))
      raises(ValueError, lambda: primepi(zoo))
      raises(ValueError, lambda: primepi(nan))
  
  
  def test_composite():
      from sympy.ntheory.generate import sieve
      sieve._reset()
      assert composite(1) == 4
      assert composite(2) == 6
      assert composite(5) == 10
      assert composite(11) == 20
      assert composite(41) == 58
      assert composite(57) == 80
      assert composite(296) == 370
      assert composite(559) == 684
      assert composite(3000) == 3488
      assert composite(4096) == 4736
      assert composite(9096) == 10368
      assert composite(25023) == 28088
      sieve.extend(3000)
      assert composite(1957) == 2300
      assert composite(2568) == 2998
      raises(ValueError, lambda: composite(0))
  
  
  def test_compositepi():
      assert compositepi(1) == 0
      assert compositepi(2) == 0
      assert compositepi(5) == 1
      assert compositepi(11) == 5
      assert compositepi(57) == 40
      assert compositepi(296) == 233
      assert compositepi(559) == 456
      assert compositepi(3000) == 2569
      assert compositepi(4096) == 3531
      assert compositepi(9096) == 7967
      assert compositepi(25023) == 22259
      assert compositepi(10**8) == 94238544
      assert compositepi(253425253) == 239568856
      assert compositepi(8769575643) == 8368111320
      sieve.extend(3000)
      assert compositepi(2321) == 1976
  
  
  def test_generate():
      from sympy.ntheory.generate import sieve
      sieve._reset()
      assert nextprime(-4) == 2
      assert nextprime(2) == 3
      assert nextprime(5) == 7
      assert nextprime(12) == 13
      assert prevprime(3) == 2
      assert prevprime(7) == 5
      assert prevprime(13) == 11
      assert prevprime(19) == 17
      assert prevprime(20) == 19
  
      sieve.extend_to_no(9)
      assert sieve._list[-1] == 23
  
      assert sieve._list[-1] < 31
      assert 31 in sieve
  
      assert nextprime(90) == 97
      assert nextprime(10**40) == (10**40 + 121)
      assert prevprime(97) == 89
      assert prevprime(10**40) == (10**40 - 17)
  
      assert list(sieve.primerange(10, 1)) == []
      assert list(sieve.primerange(5, 9)) == [5, 7]
      sieve._reset(prime=True)
      assert list(sieve.primerange(2, 12)) == [2, 3, 5, 7, 11]
  
      assert list(sieve.totientrange(5, 15)) == [4, 2, 6, 4, 6, 4, 10, 4, 12, 6]
      sieve._reset(totient=True)
      assert list(sieve.totientrange(3, 13)) == [2, 2, 4, 2, 6, 4, 6, 4, 10, 4]
      assert list(sieve.totientrange(900, 1000)) == [totient(x) for x in range(900, 1000)]
      assert list(sieve.totientrange(0, 1)) == []
      assert list(sieve.totientrange(1, 2)) == [1]
  
      assert list(sieve.mobiusrange(5, 15)) == [-1, 1, -1, 0, 0, 1, -1, 0, -1, 1]
      sieve._reset(mobius=True)
      assert list(sieve.mobiusrange(3, 13)) == [-1, 0, -1, 1, -1, 0, 0, 1, -1, 0]
      assert list(sieve.mobiusrange(1050, 1100)) == [mobius(x) for x in range(1050, 1100)]
      assert list(sieve.mobiusrange(0, 1)) == []
      assert list(sieve.mobiusrange(1, 2)) == [1]
  
      assert list(primerange(10, 1)) == []
      assert list(primerange(2, 7)) == [2, 3, 5]
      assert list(primerange(2, 10)) == [2, 3, 5, 7]
      assert list(primerange(1050, 1100)) == [1051, 1061,
          1063, 1069, 1087, 1091, 1093, 1097]
      s = Sieve()
      for i in range(30, 2350, 376):
          for j in range(2, 5096, 1139):
              A = list(s.primerange(i, i + j))
              B = list(primerange(i, i + j))
              assert A == B
      s = Sieve()
      assert s[10] == 29
  
      assert nextprime(2, 2) == 5
  
      raises(ValueError, lambda: totient(0))
  
      raises(ValueError, lambda: reduced_totient(0))
  
      raises(ValueError, lambda: primorial(0))
  
      assert mr(1, [2]) is False
  
      func = lambda i: (i**2 + 1) % 51
      assert next(cycle_length(func, 4)) == (6, 2)
      assert list(cycle_length(func, 4, values=True)) == \
          [17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14]
      assert next(cycle_length(func, 4, nmax=5)) == (5, None)
      assert list(cycle_length(func, 4, nmax=5, values=True)) == \
          [17, 35, 2, 5, 26]
      sieve.extend(3000)
      assert nextprime(2968) == 2969
      assert prevprime(2930) == 2927
      raises(ValueError, lambda: prevprime(1))
  
  
  def test_randprime():
      assert randprime(10, 1) is None
      assert randprime(2, 3) == 2
      assert randprime(1, 3) == 2
      assert randprime(3, 5) == 3
      raises(ValueError, lambda: randprime(20, 22))
      for a in [100, 300, 500, 250000]:
          for b in [100, 300, 500, 250000]:
              p = randprime(a, a + b)
              assert a <= p < (a + b) and isprime(p)
  
  
  def test_primorial():
      assert primorial(1) == 2
      assert primorial(1, nth=0) == 1
      assert primorial(2) == 6
      assert primorial(2, nth=0) == 2
      assert primorial(4, nth=0) == 6
  
  
  def test_search():
      assert 2 in sieve
      assert 2.1 not in sieve
      assert 1 not in sieve
      assert 2**1000 not in sieve
      raises(ValueError, lambda: sieve.search(1))
  
  
  def test_sieve_slice():
      assert sieve[5] == 11
      assert list(sieve[5:10]) == [sieve[x] for x in range(5, 10)]
      assert list(sieve[5:10:2]) == [sieve[x] for x in range(5, 10, 2)]
  