number.py

#
#   number.py : Number-theoretic functions
#
#  Part of the Python Cryptography Toolkit
#
#  Written by Andrew M. Kuchling, Barry A. Warsaw, and others
#
# ===================================================================
# The contents of this file are dedicated to the public domain.  To
# the extent that dedication to the public domain is not available,
# everyone is granted a worldwide, perpetual, royalty-free,
# non-exclusive license to exercise all rights associated with the
# contents of this file for any purpose whatsoever.
# No rights are reserved.
#
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
# EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
# MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
# NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS
# BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN
# ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
# CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
# SOFTWARE.
# ===================================================================
#

__revision__ = "$Id$"

from Crypto.pct_warnings import GetRandomNumber_DeprecationWarning, PowmInsecureWarning
from warnings import warn as _warn
import math
import sys
from Crypto.Util.py3compat import *

bignum = long
try:
    from Crypto.PublicKey import _fastmath
except ImportError:
    # For production, we are going to let import issues due to gmp/mpir shared
    # libraries not loading slide silently and use slowmath. If you'd rather
    # see an exception raised if _fastmath exists but cannot be imported,
    # uncomment the below
    #
    # from distutils.sysconfig import get_config_var
    # import inspect, os
    # _fm_path = os.path.normpath(os.path.dirname(os.path.abspath(
        # inspect.getfile(inspect.currentframe())))
        # +"/../../PublicKey/_fastmath"+get_config_var("SO"))
    # if os.path.exists(_fm_path):
        # raise ImportError("While the _fastmath module exists, importing "+
            # "it failed. This may point to the gmp or mpir shared library "+
            # "not being in the path. _fastmath was found at "+_fm_path)
    _fastmath = None

# You need libgmp v5 or later to get mpz_powm_sec.  Warn if it's not available.
if _fastmath is not None and not _fastmath.HAVE_DECL_MPZ_POWM_SEC:
    _warn("Not using mpz_powm_sec.  You should rebuild using libgmp >= 5 to avoid timing attack vulnerability.", PowmInsecureWarning)

# New functions
from _number_new import *

# Commented out and replaced with faster versions below
## def long2str(n):
##     s=''
##     while n>0:
##         s=chr(n & 255)+s
##         n=n>>8
##     return s

## import types
## def str2long(s):
##     if type(s)!=types.StringType: return s   # Integers will be left alone
##     return reduce(lambda x,y : x*256+ord(y), s, 0L)

def size (N):
    """size(N:long) : int
    Returns the size of the number N in bits.
    """
    bits = 0
    while N >> bits:
        bits += 1
    return bits

def getRandomNumber(N, randfunc=None):
    """Deprecated.  Use getRandomInteger or getRandomNBitInteger instead."""
    warnings.warn("Crypto.Util.number.getRandomNumber has confusing semantics"+
    "and has been deprecated.  Use getRandomInteger or getRandomNBitInteger instead.",
        GetRandomNumber_DeprecationWarning)
    return getRandomNBitInteger(N, randfunc)

def getRandomInteger(N, randfunc=None):
    """getRandomInteger(N:int, randfunc:callable):long
    Return a random number with at most N bits.

    If randfunc is omitted, then Random.new().read is used.

    This function is for internal use only and may be renamed or removed in
    the future.
    """
    if randfunc is None:
        _import_Random()
        randfunc = Random.new().read

    S = randfunc(N>>3)
    odd_bits = N % 8
    if odd_bits != 0:
        char = ord(randfunc(1)) >> (8-odd_bits)
        S = bchr(char) + S
    value = bytes_to_long(S)
    return value

def getRandomRange(a, b, randfunc=None):
    """getRandomRange(a:int, b:int, randfunc:callable):long
    Return a random number n so that a <= n < b.

    If randfunc is omitted, then Random.new().read is used.

    This function is for internal use only and may be renamed or removed in
    the future.
    """
    range_ = b - a - 1
    bits = size(range_)
    value = getRandomInteger(bits, randfunc)
    while value > range_:
        value = getRandomInteger(bits, randfunc)
    return a + value

def getRandomNBitInteger(N, randfunc=None):
    """getRandomInteger(N:int, randfunc:callable):long
    Return a random number with exactly N-bits, i.e. a random number
    between 2**(N-1) and (2**N)-1.

    If randfunc is omitted, then Random.new().read is used.

    This function is for internal use only and may be renamed or removed in
    the future.
    """
    value = getRandomInteger (N-1, randfunc)
    value |= 2L ** (N-1)                # Ensure high bit is set
    assert size(value) >= N
    return value

def GCD(x,y):
    """GCD(x:long, y:long): long
    Return the GCD of x and y.
    """
    x = abs(x) ; y = abs(y)
    while x > 0:
        x, y = y % x, x
    return y

def inverse(u, v):
    """inverse(u:long, v:long):long
    Return the inverse of u mod v.
    """
    u3, v3 = long(u), long(v)
    u1, v1 = 1L, 0L
    while v3 > 0:
        q=divmod(u3, v3)[0]
        u1, v1 = v1, u1 - v1*q
        u3, v3 = v3, u3 - v3*q
    while u1<0:
        u1 = u1 + v
    return u1

# Given a number of bits to generate and a random generation function,
# find a prime number of the appropriate size.

def getPrime(N, randfunc=None):
    """getPrime(N:int, randfunc:callable):long
    Return a random N-bit prime number.

    If randfunc is omitted, then Random.new().read is used.
    """
    if randfunc is None:
        _import_Random()
        randfunc = Random.new().read

    number=getRandomNBitInteger(N, randfunc) | 1
    while (not isPrime(number, randfunc=randfunc)):
        number=number+2
    return number


def _rabinMillerTest(n, rounds, randfunc=None):
    """_rabinMillerTest(n:long, rounds:int, randfunc:callable):int
    Tests if n is prime.
    Returns 0 when n is definitly composite.
    Returns 1 when n is probably prime.
    Returns 2 when n is definitly prime.

    If randfunc is omitted, then Random.new().read is used.

    This function is for internal use only and may be renamed or removed in
    the future.
    """
    # check special cases (n==2, n even, n < 2)
    if n < 3 or (n & 1) == 0:
        return n == 2
    # n might be very large so it might be beneficial to precalculate n-1
    n_1 = n - 1
    # determine m and b so that 2**b * m = n - 1 and b maximal
    b = 0
    m = n_1
    while (m & 1) == 0:
        b += 1
        m >>= 1

    tested = []
    # we need to do at most n-2 rounds.
    for i in xrange (min (rounds, n-2)):
        # randomly choose a < n and make sure it hasn't been tested yet
        a = getRandomRange (2, n, randfunc)
        while a in tested:
            a = getRandomRange (2, n, randfunc)
        tested.append (a)
        # do the rabin-miller test
        z = pow (a, m, n) # (a**m) % n
        if z == 1 or z == n_1:
            continue
        composite = 1
        for r in xrange (b):
            z = (z * z) % n
            if z == 1:
                return 0
            elif z == n_1:
                composite = 0
                break
        if composite:
            return 0
    return 1

def getStrongPrime(N, e=0, false_positive_prob=1e-6, randfunc=None):
    """getStrongPrime(N:int, e:int, false_positive_prob:float, randfunc:callable):long
    Return a random strong N-bit prime number.
    In this context p is a strong prime if p-1 and p+1 have at
    least one large prime factor.
    N should be a multiple of 128 and > 512.

    If e is provided the returned prime p-1 will be coprime to e
    and thus suitable for RSA where e is the public exponent.

    The optional false_positive_prob is the statistical probability
    that true is returned even though it is not (pseudo-prime).
    It defaults to 1e-6 (less than 1:1000000).
    Note that the real probability of a false-positive is far less. This is
    just the mathematically provable limit.

    randfunc should take a single int parameter and return that
    many random bytes as a string.
    If randfunc is omitted, then Random.new().read is used.
    """
    # This function was implemented following the
    # instructions found in the paper:
    #   "FAST GENERATION OF RANDOM, STRONG RSA PRIMES"
    #   by Robert D. Silverman
    #   RSA Laboratories
    #   May 17, 1997
    # which by the time of writing could be freely downloaded here:
    # http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.17.2713&rep=rep1&type=pdf

    # Use the accelerator if available
    if _fastmath is not None:
        return _fastmath.getStrongPrime(long(N), long(e), false_positive_prob,
            randfunc)

    if (N < 512) or ((N % 128) != 0):
        raise ValueError ("bits must be multiple of 128 and > 512")

    rabin_miller_rounds = int(math.ceil(-math.log(false_positive_prob)/math.log(4)))

    # calculate range for X
    #   lower_bound = sqrt(2) * 2^{511 + 128*x}
    #   upper_bound = 2^{512 + 128*x} - 1
    x = (N - 512) >> 7;
    # We need to approximate the sqrt(2) in the lower_bound by an integer
    # expression because floating point math overflows with these numbers
    lower_bound = divmod(14142135623730950489L * (2L ** (511 + 128*x)),
                         10000000000000000000L)[0]
    upper_bound = (1L << (512 + 128*x)) - 1
    # Randomly choose X in calculated range
    X = getRandomRange (lower_bound, upper_bound, randfunc)

    # generate p1 and p2
    p = [0, 0]
    for i in (0, 1):
        # randomly choose 101-bit y
        y = getRandomNBitInteger (101, randfunc)
        # initialize the field for sieving
        field = [0] * 5 * len (sieve_base)
        # sieve the field
        for prime in sieve_base:
            offset = y % prime
            for j in xrange ((prime - offset) % prime, len (field), prime):
                field[j] = 1

        # look for suitable p[i] starting at y
        result = 0
        for j in range(len(field)):
            composite = field[j]
            # look for next canidate
            if composite:
                continue
            tmp = y + j
            result = _rabinMillerTest (tmp, rabin_miller_rounds)
            if result > 0:
                p[i] = tmp
                break
        if result == 0:
            raise RuntimeError ("Couln't find prime in field. "
                                "Developer: Increase field_size")

    # Calculate R
    #     R = (p2^{-1} mod p1) * p2 - (p1^{-1} mod p2) * p1
    tmp1 = inverse (p[1], p[0]) * p[1]  # (p2^-1 mod p1)*p2
    tmp2 = inverse (p[0], p[1]) * p[0]  # (p1^-1 mod p2)*p1
    R = tmp1 - tmp2 # (p2^-1 mod p1)*p2 - (p1^-1 mod p2)*p1

    # search for final prime number starting by Y0
    #    Y0 = X + (R - X mod p1p2)
    increment = p[0] * p[1]
    X = X + (R - (X % increment))
    while 1:
        is_possible_prime = 1
        # first check candidate against sieve_base
        for prime in sieve_base:
            if (X % prime) == 0:
                is_possible_prime = 0
                break
        # if e is given make sure that e and X-1 are coprime
        # this is not necessarily a strong prime criterion but useful when
        # creating them for RSA where the p-1 and q-1 should be coprime to
        # the public exponent e
        if e and is_possible_prime:
            if e & 1:
                if GCD (e, X-1) != 1:
                    is_possible_prime = 0
            else:
                if GCD (e, divmod((X-1),2)[0]) != 1:
                    is_possible_prime = 0

        # do some Rabin-Miller-Tests
        if is_possible_prime:
            result = _rabinMillerTest (X, rabin_miller_rounds)
            if result > 0:
                break
        X += increment
		# abort when X has more bits than requested
		# TODO: maybe we shouldn't abort but rather start over.
        if X >= 1L << N:
            raise RuntimeError ("Couln't find prime in field. "
                                "Developer: Increase field_size")
    return X

def isPrime(N, false_positive_prob=1e-6, randfunc=None):
    """isPrime(N:long, false_positive_prob:float, randfunc:callable):bool
    Return true if N is prime.

    The optional false_positive_prob is the statistical probability
    that true is returned even though it is not (pseudo-prime).
    It defaults to 1e-6 (less than 1:1000000).
    Note that the real probability of a false-positive is far less. This is
    just the mathematically provable limit.

    If randfunc is omitted, then Random.new().read is used.
    """
    if _fastmath is not None:
        return _fastmath.isPrime(long(N), false_positive_prob, randfunc)

    if N < 3 or N & 1 == 0:
        return N == 2
    for p in sieve_base:
        if N == p:
            return 1
        if N % p == 0:
            return 0

    rounds = int(math.ceil(-math.log(false_positive_prob)/math.log(4)))
    return _rabinMillerTest(N, rounds, randfunc)


# Improved conversion functions contributed by Barry Warsaw, after
# careful benchmarking

import struct

def long_to_bytes(n, blocksize=0):
    """long_to_bytes(n:long, blocksize:int) : string
    Convert a long integer to a byte string.

    If optional blocksize is given and greater than zero, pad the front of the
    byte string with binary zeros so that the length is a multiple of
    blocksize.
    """
    # after much testing, this algorithm was deemed to be the fastest
    s = b('')
    n = long(n)
    pack = struct.pack
    while n > 0:
        s = pack('>I', n & 0xffffffffL) + s
        n = n >> 32
    # strip off leading zeros
    for i in range(len(s)):
        if s[i] != b('\000')[0]:
            break
    else:
        # only happens when n == 0
        s = b('\000')
        i = 0
    s = s[i:]
    # add back some pad bytes.  this could be done more efficiently w.r.t. the
    # de-padding being done above, but sigh...
    if blocksize > 0 and len(s) % blocksize:
        s = (blocksize - len(s) % blocksize) * b('\000') + s
    return s

def bytes_to_long(s):
    """bytes_to_long(string) : long
    Convert a byte string to a long integer.

    This is (essentially) the inverse of long_to_bytes().
    """
    acc = 0L
    unpack = struct.unpack
    length = len(s)
    if length % 4:
        extra = (4 - length % 4)
        s = b('\000') * extra + s
        length = length + extra
    for i in range(0, length, 4):
        acc = (acc << 32) + unpack('>I', s[i:i+4])[0]
    return acc

# For backwards compatibility...
import warnings
def long2str(n, blocksize=0):
    warnings.warn("long2str() has been replaced by long_to_bytes()")
    return long_to_bytes(n, blocksize)
def str2long(s):
    warnings.warn("str2long() has been replaced by bytes_to_long()")
    return bytes_to_long(s)

def _import_Random():
    # This is called in a function instead of at the module level in order to
    # avoid problems with recursive imports
    global Random, StrongRandom
    from Crypto import Random
    from Crypto.Random.random import StrongRandom



# The first 10000 primes used for checking primality.
# This should be enough to eliminate most of the odd
# numbers before needing to do a Rabin-Miller test at all.
sieve_base = (
     2,      3,      5,      7,     11,     13,     17,     19,     23,     29,
    31,     37,     41,     43,     47,     53,     59,     61,     67,     71,
    73,     79,     83,     89,     97,    101,    103,    107,    109,    113,
   127,    131,    137,    139,    149,    151,    157,    163,    167,    173,
   179,    181,    191,    193,    197,    199,    211,    223,    227,    229,
   233,    239,    241,    251,    257,    263,    269,    271,    277,    281,
   283,    293,    307,    311,    313,    317,    331,    337,    347,    349,
   353,    359,    367,    373,    379,    383,    389,    397,    401,    409,
   419,    421,    431,    433,    439,    443,    449,    457,    461,    463,
   467,    479,    487,    491,    499,    503,    509,    521,    523,    541,
   547,    557,    563,    569,    571,    577,    587,    593,    599,    601,
   607,    613,    617,    619,    631,    641,    643,    647,    653,    659,
   661,    673,    677,    683,    691,    701,    709,    719,    727,    733,
   739,    743,    751,    757,    761,    769,    773,    787,    797,    809,
   811,    821,    823,    827,    829,    839,    853,    857,    859,    863,
   877,    881,    883,    887,    907,    911,    919,    929,    937,    941,
   947,    953,    967,    971,    977,    983,    991,    997,   1009,   1013,
  1019,   1021,   1031,   1033,   1039,   1049,   1051,   1061,   1063,   1069,
  1087,   1091,   1093,   1097,   1103,   1109,   1117,   1123,   1129,   1151,
  1153,   1163,   1171,   1181,   1187,   1193,   1201,   1213,   1217,   1223,
  1229,   1231,   1237,   1249,   1259,   1277,   1279,   1283,   1289,   1291,
  1297,   1301,   1303,   1307,   1319,   1321,   1327,   1361,   1367,   1373,
  1381,   1399,   1409,   1423,   1427,   1429,   1433,   1439,   1447,   1451,
  1453,   1459,   1471,   1481,   1483,   1487,   1489,   1493,   1499,   1511,
  1523,   1531,   1543,   1549,   1553,   1559,   1567,   1571,   1579,   1583,
  1597,   1601,   1607,   1609,   1613,   1619,   1621,   1627,   1637,   1657,
  1663,   1667,   1669,   1693,   1697,   1699,   1709,   1721,   1723,   1733,
  1741,   1747,   1753,   1759,   1777,   1783,   1787,   1789,   1801,   1811,
  1823,   1831,   1847,   1861,   1867,   1871,   1873,   1877,   1879,   1889,
  1901,   1907,   1913,   1931,   1933,   1949,   1951,   1973,   1979,   1987,
  1993,   1997,   1999,   2003,   2011,   2017,   2027,   2029,   2039,   2053,
  2063,   2069,   2081,   2083,   2087,   2089,   2099,   2111,   2113,   2129,
  2131,   2137,   2141,   2143,   2153,   2161,   2179,   2203,   2207,   2213,
  2221,   2237,   2239,   2243,   2251,   2267,   2269,   2273,   2281,   2287,
  2293,   2297,   2309,   2311,   2333,   2339,   2341,   2347,   2351,   2357,
  2371,   2377,   2381,   2383,   2389,   2393,   2399,   2411,   2417,   2423,
  2437,   2441,   2447,   2459,   2467,   2473,   2477,   2503,   2521,   2531,
  2539,   2543,   2549,   2551,   2557,   2579,   2591,   2593,   2609,   2617,
  2621,   2633,   2647,   2657,   2659,   2663,   2671,   2677,   2683,   2687,
  2689,   2693,   2699,   2707,   2711,   2713,   2719,   2729,   2731,   2741,
  2749,   2753,   2767,   2777,   2789,   2791,   2797,   2801,   2803,   2819,
  2833,   2837,   2843,   2851,   2857,   2861,   2879,   2887,   2897,   2903,
  2909,   2917,   2927,   2939,   2953,   2957,   2963,   2969,   2971,   2999,
  3001,   3011,   3019,   3023,   3037,   3041,   3049,   3061,   3067,   3079,
  3083,   3089,   3109,   3119,   3121,   3137,   3163,   3167,   3169,   3181,
  3187,   3191,   3203,   3209,   3217,   3221,   3229,   3251,   3253,   3257,
  3259,   3271,   3299,   3301,   3307,   3313,   3319,   3323,   3329,   3331,
  3343,   3347,   3359,   3361,   3371,   3373,   3389,   3391,   3407,   3413,
  3433,   3449,   3457,   3461,   3463,   3467,   3469,   3491,   3499,   3511,
  3517,   3527,   3529,   3533,   3539,   3541,   3547,   3557,   3559,   3571,
  3581,   3583,   3593,   3607,   3613,   3617,   3623,   3631,   3637,   3643,
  3659,   3671,   3673,   3677,   3691,   3697,   3701,   3709,   3719,   3727,
  3733,   3739,   3761,   3767,   3769,   3779,   3793,   3797,   3803,   3821,
  3823,   3833,   3847,   3851,   3853,   3863,   3877,   3881,   3889,   3907,
  3911,   3917,   3919,   3923,   3929,   3931,   3943,   3947,   3967,   3989,
  4001,   4003,   4007,   4013,   4019,   4021,   4027,   4049,   4051,   4057,
  4073,   4079,   4091,   4093,   4099,   4111,   4127,   4129,   4133,   4139,
  4153,   4157,   4159,   4177,   4201,   4211,   4217,   4219,   4229,   4231,
  4241,   4243,   4253,   4259,   4261,   4271,   4273,   4283,   4289,   4297,
  4327,   4337,   4339,   4349,   4357,   4363,   4373,   4391,   4397,   4409,
  4421,   4423,   4441,   4447,   4451,   4457,   4463,   4481,   4483,   4493,
  4507,   4513,   4517,   4519,   4523,   4547,   4549,   4561,   4567,   4583,
  4591,   4597,   4603,   4621,   4637,   4639,   4643,   4649,   4651,   4657,
  4663,   4673,   4679,   4691,   4703,   4721,   4723,   4729,   4733,   4751,
  4759,   4783,   4787,   4789,   4793,   4799,   4801,   4813,   4817,   4831,
  4861,   4871,   4877,   4889,   4903,   4909,   4919,   4931,   4933,   4937,
  4943,   4951,   4957,   4967,   4969,   4973,   4987,   4993,   4999,   5003,
  5009,   5011,   5021,   5023,   5039,   5051,   5059,   5077,   5081,   5087,
  5099,   5101,   5107,   5113,   5119,   5147,   5153,   5167,   5171,   5179,
  5189,   5197,   5209,   5227,   5231,   5233,   5237,   5261,   5273,   5279,
  5281,   5297,   5303,   5309,   5323,   5333,   5347,   5351,   5381,   5387,
  5393,   5399,   5407,   5413,   5417,   5419,   5431,   5437,   5441,   5443,
  5449,   5471,   5477,   5479,   5483,   5501,   5503,   5507,   5519,   5521,
  5527,   5531,   5557,   5563,   5569,   5573,   5581,   5591,   5623,   5639,
  5641,   5647,   5651,   5653,   5657,   5659,   5669,   5683,   5689,   5693,
  5701,   5711,   5717,   5737,   5741,   5743,   5749,   5779,   5783,   5791,
  5801,   5807,   5813,   5821,   5827,   5839,   5843,   5849,   5851,   5857,
  5861,   5867,   5869,   5879,   5881,   5897,   5903,   5923,   5927,   5939,
  5953,   5981,   5987,   6007,   6011,   6029,   6037,   6043,   6047,   6053,
  6067,   6073,   6079,   6089,   6091,   6101,   6113,   6121,   6131,   6133,
  6143,   6151,   6163,   6173,   6197,   6199,   6203,   6211,   6217,   6221,
  6229,   6247,   6257,   6263,   6269,   6271,   6277,   6287,   6299,   6301,
  6311,   6317,   6323,   6329,   6337,   6343,   6353,   6359,   6361,   6367,
  6373,   6379,   6389,   6397,   6421,   6427,   6449,   6451,   6469,   6473,
  6481,   6491,   6521,   6529,   6547,   6551,   6553,   6563,   6569,   6571,
  6577,   6581,   6599,   6607,   6619,   6637,   6653,   6659,   6661,   6673,
  6679,   6689,   6691,   6701,   6703,   6709,   6719,   6733,   6737,   6761,
  6763,   6779,   6781,   6791,   6793,   6803,   6823,   6827,   6829,   6833,
  6841,   6857,   6863,   6869,   6871,   6883,   6899,   6907,   6911,   6917,
  6947,   6949,   6959,   6961,   6967,   6971,   6977,   6983,   6991,   6997,
  7001,   7013,   7019,   7027,   7039,   7043,   7057,   7069,   7079,   7103,
  7109,   7121,   7127,   7129,   7151,   7159,   7177,   7187,   7193,   7207,
  7211,   7213,   7219,   7229,   7237,   7243,   7247,   7253,   7283,   7297,
  7307,   7309,   7321,   7331,   7333,   7349,   7351,   7369,   7393,   7411,
  7417,   7433,   7451,   7457,   7459,   7477,   7481,   7487,   7489,   7499,
  7507,   7517,   7523,   7529,   7537,   7541,   7547,   7549,   7559,   7561,
  7573,   7577,   7583,   7589,   7591,   7603,   7607,   7621,   7639,   7643,
  7649,   7669,   7673,   7681,   7687,   7691,   7699,   7703,   7717,   7723,
  7727,   7741,   7753,   7757,   7759,   7789,   7793,   7817,   7823,   7829,
  7841,   7853,   7867,   7873,   7877,   7879,   7883,   7901,   7907,   7919,
  7927,   7933,   7937,   7949,   7951,   7963,   7993,   8009,   8011,   8017,
  8039,   8053,   8059,   8069,   8081,   8087,   8089,   8093,   8101,   8111,
  8117,   8123,   8147,   8161,   8167,   8171,   8179,   8191,   8209,   8219,
  8221,   8231,   8233,   8237,   8243,   8263,   8269,   8273,   8287,   8291,
  8293,   8297,   8311,   8317,   8329,   8353,   8363,   8369,   8377,   8387,
  8389,   8419,   8423,   8429,   8431,   8443,   8447,   8461,   8467,   8501,
  8513,   8521,   8527,   8537,   8539,   8543,   8563,   8573,   8581,   8597,
  8599,   8609,   8623,   8627,   8629,   8641,   8647,   8663,   8669,   8677,
  8681,   8689,   8693,   8699,   8707,   8713,   8719,   8731,   8737,   8741,
  8747,   8753,   8761,   8779,   8783,   8803,   8807,   8819,   8821,   8831,
  8837,   8839,   8849,   8861,   8863,   8867,   8887,   8893,   8923,   8929,
  8933,   8941,   8951,   8963,   8969,   8971,   8999,   9001,   9007,   9011,
  9013,   9029,   9041,   9043,   9049,   9059,   9067,   9091,   9103,   9109,
  9127,   9133,   9137,   9151,   9157,   9161,   9173,   9181,   9187,   9199,
  9203,   9209,   9221,   9227,   9239,   9241,   9257,   9277,   9281,   9283,
  9293,   9311,   9319,   9323,   9337,   9341,   9343,   9349,   9371,   9377,
  9391,   9397,   9403,   9413,   9419,   9421,   9431,   9433,   9437,   9439,
  9461,   9463,   9467,   9473,   9479,   9491,   9497,   9511,   9521,   9533,
  9539,   9547,   9551,   9587,   9601,   9613,   9619,   9623,   9629,   9631,
  9643,   9649,   9661,   9677,   9679,   9689,   9697,   9719,   9721,   9733,
  9739,   9743,   9749,   9767,   9769,   9781,   9787,   9791,   9803,   9811,
  9817,   9829,   9833,   9839,   9851,   9857,   9859,   9871,   9883,   9887,
  9901,   9907,   9923,   9929,   9931,   9941,   9949,   9967,   9973,  10007,
 10009,  10037,  10039,  10061,  10067,  10069,  10079,  10091,  10093,  10099,
 10103,  10111,  10133,  10139,  10141,  10151,  10159,  10163,  10169,  10177,
 10181,  10193,  10211,  10223,  10243,  10247,  10253,  10259,  10267,  10271,
 10273,  10289,  10301,  10303,  10313,  10321,  10331,  10333,  10337,  10343,
 10357,  10369,  10391,  10399,  10427,  10429,  10433,  10453,  10457,  10459,
 10463,  10477,  10487,  10499,  10501,  10513,  10529,  10531,  10559,  10567,
 10589,  10597,  10601,  10607,  10613,  10627,  10631,  10639,  10651,  10657,
 10663,  10667,  10687,  10691,  10709,  10711,  10723,  10729,  10733,  10739,
 10753,  10771,  10781,  10789,  10799,  10831,  10837,  10847,  10853,  10859,
 10861,  10867,  10883,  10889,  10891,  10903,  10909,  10937,  10939,  10949,
 10957,  10973,  10979,  10987,  10993,  11003,  11027,  11047,  11057,  11059,
 11069,  11071,  11083,  11087,  11093,  11113,  11117,  11119,  11131,  11149,
 11159,  11161,  11171,  11173,  11177,  11197,  11213,  11239,  11243,  11251,
 11257,  11261,  11273,  11279,  11287,  11299,  11311,  11317,  11321,  11329,
 11351,  11353,  11369,  11383,  11393,  11399,  11411,  11423,  11437,  11443,
 11447,  11467,  11471,  11483,  11489,  11491,  11497,  11503,  11519,  11527,
 11549,  11551,  11579,  11587,  11593,  11597,  11617,  11621,  11633,  11657,
 11677,  11681,  11689,  11699,  11701,  11717,  11719,  11731,  11743,  11777,
 11779,  11783,  11789,  11801,  11807,  11813,  11821,  11827,  11831,  11833,
 11839,  11863,  11867,  11887,  11897,  11903,  11909,  11923,  11927,  11933,
 11939,  11941,  11953,  11959,  11969,  11971,  11981,  11987,  12007,  12011,
 12037,  12041,  12043,  12049,  12071,  12073,  12097,  12101,  12107,  12109,
 12113,  12119,  12143,  12149,  12157,  12161,  12163,  12197,  12203,  12211,
 12227,  12239,  12241,  12251,  12253,  12263,  12269,  12277,  12281,  12289,
 12301,  12323,  12329,  12343,  12347,  12373,  12377,  12379,  12391,  12401,
 12409,  12413,  12421,  12433,  12437,  12451,  12457,  12473,  12479,  12487,
 12491,  12497,  12503,  12511,  12517,  12527,  12539,  12541,  12547,  12553,
 12569,  12577,  12583,  12589,  12601,  12611,  12613,  12619,  12637,  12641,
 12647,  12653,  12659,  12671,  12689,  12697,  12703,  12713,  12721,  12739,
 12743,  12757,  12763,  12781,  12791,  12799,  12809,  12821,  12823,  12829,
 12841,  12853,  12889,  12893,  12899,  12907,  12911,  12917,  12919,  12923,
 12941,  12953,  12959,  12967,  12973,  12979,  12983,  13001,  13003,  13007,
 13009,  13033,  13037,  13043,  13049,  13063,  13093,  13099,  13103,  13109,
 13121,  13127,  13147,  13151,  13159,  13163,  13171,  13177,  13183,  13187,
 13217,  13219,  13229,  13241,  13249,  13259,  13267,  13291,  13297,  13309,
 13313,  13327,  13331,  13337,  13339,  13367,  13381,  13397,  13399,  13411,
 13417,  13421,  13441,  13451,  13457,  13463,  13469,  13477,  13487,  13499,
 13513,  13523,  13537,  13553,  13567,  13577,  13591,  13597,  13613,  13619,
 13627,  13633,  13649,  13669,  13679,  13681,  13687,  13691,  13693,  13697,
 13709,  13711,  13721,  13723,  13729,  13751,  13757,  13759,  13763,  13781,
 13789,  13799,  13807,  13829,  13831,  13841,  13859,  13873,  13877,  13879,
 13883,  13901,  13903,  13907,  13913,  13921,  13931,  13933,  13963,  13967,
 13997,  13999,  14009,  14011,  14029,  14033,  14051,  14057,  14071,  14081,
 14083,  14087,  14107,  14143,  14149,  14153,  14159,  14173,  14177,  14197,
 14207,  14221,  14243,  14249,  14251,  14281,  14293,  14303,  14321,  14323,
 14327,  14341,  14347,  14369,  14387,  14389,  14401,  14407,  14411,  14419,
 14423,  14431,  14437,  14447,  14449,  14461,  14479,  14489,  14503,  14519,
 14533,  14537,  14543,  14549,  14551,  14557,  14561,  14563,  14591,  14593,
 14621,  14627,  14629,  14633,  14639,  14653,  14657,  14669,  14683,  14699,
 14713,  14717,  14723,  14731,  14737,  14741,  14747,  14753,  14759,  14767,
 14771,  14779,  14783,  14797,  14813,  14821,  14827,  14831,  14843,  14851,
 14867,  14869,  14879,  14887,  14891,  14897,  14923,  14929,  14939,  14947,
 14951,  14957,  14969,  14983,  15013,  15017,  15031,  15053,  15061,  15073,
 15077,  15083,  15091,  15101,  15107,  15121,  15131,  15137,  15139,  15149,
 15161,  15173,  15187,  15193,  15199,  15217,  15227,  15233,  15241,  15259,
 15263,  15269,  15271,  15277,  15287,  15289,  15299,  15307,  15313,  15319,
 15329,  15331,  15349,  15359,  15361,  15373,  15377,  15383,  15391,  15401,
 15413,  15427,  15439,  15443,  15451,  15461,  15467,  15473,  15493,  15497,
 15511,  15527,  15541,  15551,  15559,  15569,  15581,  15583,  15601,  15607,
 15619,  15629,  15641,  15643,  15647,  15649,  15661,  15667,  15671,  15679,
 15683,  15727,  15731,  15733,  15737,  15739,  15749,  15761,  15767,  15773,
 15787,  15791,  15797,  15803,  15809,  15817,  15823,  15859,  15877,  15881,
 15887,  15889,  15901,  15907,  15913,  15919,  15923,  15937,  15959,  15971,
 15973,  15991,  16001,  16007,  16033,  16057,  16061,  16063,  16067,  16069,
 16073,  16087,  16091,  16097,  16103,  16111,  16127,  16139,  16141,  16183,
 16187,  16189,  16193,  16217,  16223,  16229,  16231,  16249,  16253,  16267,
 16273,  16301,  16319,  16333,  16339,  16349,  16361,  16363,  16369,  16381,
 16411,  16417,  16421,  16427,  16433,  16447,  16451,  16453,  16477,  16481,
 16487,  16493,  16519,  16529,  16547,  16553,  16561,  16567,  16573,  16603,
 16607,  16619,  16631,  16633,  16649,  16651,  16657,  16661,  16673,  16691,
 16693,  16699,  16703,  16729,  16741,  16747,  16759,  16763,  16787,  16811,
 16823,  16829,  16831,  16843,  16871,  16879,  16883,  16889,  16901,  16903,
 16921,  16927,  16931,  16937,  16943,  16963,  16979,  16981,  16987,  16993,
 17011,  17021,  17027,  17029,  17033,  17041,  17047,  17053,  17077,  17093,
 17099,  17107,  17117,  17123,  17137,  17159,  17167,  17183,  17189,  17191,
 17203,  17207,  17209,  17231,  17239,  17257,  17291,  17293,  17299,  17317,
 17321,  17327,  17333,  17341,  17351,  17359,  17377,  17383,  17387,  17389,
 17393,  17401,  17417,  17419,  17431,  17443,  17449,  17467,  17471,  17477,
 17483,  17489,  17491,  17497,  17509,  17519,  17539,  17551,  17569,  17573,
 17579,  17581,  17597,  17599,  17609,  17623,  17627,  17657,  17659,  17669,
 17681,  17683,  17707,  17713,  17729,  17737,  17747,  17749,  17761,  17783,
 17789,  17791,  17807,  17827,  17837,  17839,  17851,  17863,  17881,  17891,
 17903,  17909,  17911,  17921,  17923,  17929,  17939,  17957,  17959,  17971,
 17977,  17981,  17987,  17989,  18013,  18041,  18043,  18047,  18049,  18059,
 18061,  18077,  18089,  18097,  18119,  18121,  18127,  18131,  18133,  18143,
 18149,  18169,  18181,  18191,  18199,  18211,  18217,  18223,  18229,  18233,
 18251,  18253,  18257,  18269,  18287,  18289,  18301,  18307,  18311,  18313,
 18329,  18341,  18353,  18367,  18371,  18379,  18397,  18401,  18413,  18427,
 18433,  18439,  18443,  18451,  18457,  18461,  18481,  18493,  18503,  18517,
 18521,  18523,  18539,  18541,  18553,  18583,  18587,  18593,  18617,  18637,
 18661,  18671,  18679,  18691,  18701,  18713,  18719,  18731,  18743,  18749,
 18757,  18773,  18787,  18793,  18797,  18803,  18839,  18859,  18869,  18899,
 18911,  18913,  18917,  18919,  18947,  18959,  18973,  18979,  19001,  19009,
 19013,  19031,  19037,  19051,  19069,  19073,  19079,  19081,  19087,  19121,
 19139,  19141,  19157,  19163,  19181,  19183,  19207,  19211,  19213,  19219,
 19231,  19237,  19249,  19259,  19267,  19273,  19289,  19301,  19309,  19319,
 19333,  19373,  19379,  19381,  19387,  19391,  19403,  19417,  19421,  19423,
 19427,  19429,  19433,  19441,  19447,  19457,  19463,  19469,  19471,  19477,
 19483,  19489,  19501,  19507,  19531,  19541,  19543,  19553,  19559,  19571,
 19577,  19583,  19597,  19603,  19609,  19661,  19681,  19687,  19697,  19699,
 19709,  19717,  19727,  19739,  19751,  19753,  19759,  19763,  19777,  19793,
 19801,  19813,  19819,  19841,  19843,  19853,  19861,  19867,  19889,  19891,
 19913,  19919,  19927,  19937,  19949,  19961,  19963,  19973,  19979,  19991,
 19993,  19997,  20011,  20021,  20023,  20029,  20047,  20051,  20063,  20071,
 20089,  20101,  20107,  20113,  20117,  20123,  20129,  20143,  20147,  20149,
 20161,  20173,  20177,  20183,  20201,  20219,  20231,  20233,  20249,  20261,
 20269,  20287,  20297,  20323,  20327,  20333,  20341,  20347,  20353,  20357,
 20359,  20369,  20389,  20393,  20399,  20407,  20411,  20431,  20441,  20443,
 20477,  20479,  20483,  20507,  20509,  20521,  20533,  20543,  20549,  20551,
 20563,  20593,  20599,  20611,  20627,  20639,  20641,  20663,  20681,  20693,
 20707,  20717,  20719,  20731,  20743,  20747,  20749,  20753,  20759,  20771,
 20773,  20789,  20807,  20809,  20849,  20857,  20873,  20879,  20887,  20897,
 20899,  20903,  20921,  20929,  20939,  20947,  20959,  20963,  20981,  20983,
 21001,  21011,  21013,  21017,  21019,  21023,  21031,  21059,  21061,  21067,
 21089,  21101,  21107,  21121,  21139,  21143,  21149,  21157,  21163,  21169,
 21179,  21187,  21191,  21193,  21211,  21221,  21227,  21247,  21269,  21277,
 21283,  21313,  21317,  21319,  21323,  21341,  21347,  21377,  21379,  21383,
 21391,  21397,  21401,  21407,  21419,  21433,  21467,  21481,  21487,  21491,
 21493,  21499,  21503,  21517,  21521,  21523,  21529,  21557,  21559,  21563,
 21569,  21577,  21587,  21589,  21599,  21601,  21611,  21613,  21617,  21647,
 21649,  21661,  21673,  21683,  21701,  21713,  21727,  21737,  21739,  21751,
 21757,  21767,  21773,  21787,  21799,  21803,  21817,  21821,  21839,  21841,
 21851,  21859,  21863,  21871,  21881,  21893,  21911,  21929,  21937,  21943,
 21961,  21977,  21991,  21997,  22003,  22013,  22027,  22031,  22037,  22039,
 22051,  22063,  22067,  22073,  22079,  22091,  22093,  22109,  22111,  22123,
 22129,  22133,  22147,  22153,  22157,  22159,  22171,  22189,  22193,  22229,
 22247,  22259,  22271,  22273,  22277,  22279,  22283,  22291,  22303,  22307,
 22343,  22349,  22367,  22369,  22381,  22391,  22397,  22409,  22433,  22441,
 22447,  22453,  22469,  22481,  22483,  22501,  22511,  22531,  22541,  22543,
 22549,  22567,  22571,  22573,  22613,  22619,  22621,  22637,  22639,  22643,
 22651,  22669,  22679,  22691,  22697,  22699,  22709,  22717,  22721,  22727,
 22739,  22741,  22751,  22769,  22777,  22783,  22787,  22807,  22811,  22817,
 22853,  22859,  22861,  22871,  22877,  22901,  22907,  22921,  22937,  22943,
 22961,  22963,  22973,  22993,  23003,  23011,  23017,  23021,  23027,  23029,
 23039,  23041,  23053,  23057,  23059,  23063,  23071,  23081,  23087,  23099,
 23117,  23131,  23143,  23159,  23167,  23173,  23189,  23197,  23201,  23203,
 23209,  23227,  23251,  23269,  23279,  23291,  23293,  23297,  23311,  23321,
 23327,  23333,  23339,  23357,  23369,  23371,  23399,  23417,  23431,  23447,
 23459,  23473,  23497,  23509,  23531,  23537,  23539,  23549,  23557,  23561,
 23563,  23567,  23581,  23593,  23599,  23603,  23609,  23623,  23627,  23629,
 23633,  23663,  23669,  23671,  23677,  23687,  23689,  23719,  23741,  23743,
 23747,  23753,  23761,  23767,  23773,  23789,  23801,  23813,  23819,  23827,
 23831,  23833,  23857,  23869,  23873,  23879,  23887,  23893,  23899,  23909,
 23911,  23917,  23929,  23957,  23971,  23977,  23981,  23993,  24001,  24007,
 24019,  24023,  24029,  24043,  24049,  24061,  24071,  24077,  24083,  24091,
 24097,  24103,  24107,  24109,  24113,  24121,  24133,  24137,  24151,  24169,
 24179,  24181,  24197,  24203,  24223,  24229,  24239,  24247,  24251,  24281,
 24317,  24329,  24337,  24359,  24371,  24373,  24379,  24391,  24407,  24413,
 24419,  24421,  24439,  24443,  24469,  24473,  24481,  24499,  24509,  24517,
 24527,  24533,  24547,  24551,  24571,  24593,  24611,  24623,  24631,  24659,
 24671,  24677,  24683,  24691,  24697,  24709,  24733,  24749,  24763,  24767,
 24781,  24793,  24799,  24809,  24821,  24841,  24847,  24851,  24859,  24877,
 24889,  24907,  24917,  24919,  24923,  24943,  24953,  24967,  24971,  24977,
 24979,  24989,  25013,  25031,  25033,  25037,  25057,  25073,  25087,  25097,
 25111,  25117,  25121,  25127,  25147,  25153,  25163,  25169,  25171,  25183,
 25189,  25219,  25229,  25237,  25243,  25247,  25253,  25261,  25301,  25303,
 25307,  25309,  25321,  25339,  25343,  25349,  25357,  25367,  25373,  25391,
 25409,  25411,  25423,  25439,  25447,  25453,  25457,  25463,  25469,  25471,
 25523,  25537,  25541,  25561,  25577,  25579,  25583,  25589,  25601,  25603,
 25609,  25621,  25633,  25639,  25643,  25657,  25667,  25673,  25679,  25693,
 25703,  25717,  25733,  25741,  25747,  25759,  25763,  25771,  25793,  25799,
 25801,  25819,  25841,  25847,  25849,  25867,  25873,  25889,  25903,  25913,
 25919,  25931,  25933,  25939,  25943,  25951,  25969,  25981,  25997,  25999,
 26003,  26017,  26021,  26029,  26041,  26053,  26083,  26099,  26107,  26111,
 26113,  26119,  26141,  26153,  26161,  26171,  26177,  26183,  26189,  26203,
 26209,  26227,  26237,  26249,  26251,  26261,  26263,  26267,  26293,  26297,
 26309,  26317,  26321,  26339,  26347,  26357,  26371,  26387,  26393,  26399,
 26407,  26417,  26423,  26431,  26437,  26449,  26459,  26479,  26489,  26497,
 26501,  26513,  26539,  26557,  26561,  26573,  26591,  26597,  26627,  26633,
 26641,  26647,  26669,  26681,  26683,  26687,  26693,  26699,  26701,  26711,
 26713,  26717,  26723,  26729,  26731,  26737,  26759,  26777,  26783,  26801,
 26813,  26821,  26833,  26839,  26849,  26861,  26863,  26879,  26881,  26891,
 26893,  26903,  26921,  26927,  26947,  26951,  26953,  26959,  26981,  26987,
 26993,  27011,  27017,  27031,  27043,  27059,  27061,  27067,  27073,  27077,
 27091,  27103,  27107,  27109,  27127,  27143,  27179,  27191,  27197,  27211,
 27239,  27241,  27253,  27259,  27271,  27277,  27281,  27283,  27299,  27329,
 27337,  27361,  27367,  27397,  27407,  27409,  27427,  27431,  27437,  27449,
 27457,  27479,  27481,  27487,  27509,  27527,  27529,  27539,  27541,  27551,
 27581,  27583,  27611,  27617,  27631,  27647,  27653,  27673,  27689,  27691,
 27697,  27701,  27733,  27737,  27739,  27743,  27749,  27751,  27763,  27767,
 27773,  27779,  27791,  27793,  27799,  27803,  27809,  27817,  27823,  27827,
 27847,  27851,  27883,  27893,  27901,  27917,  27919,  27941,  27943,  27947,
 27953,  27961,  27967,  27983,  27997,  28001,  28019,  28027,  28031,  28051,
 28057,  28069,  28081,  28087,  28097,  28099,  28109,  28111,  28123,  28151,
 28163,  28181,  28183,  28201,  28211,  28219,  28229,  28277,  28279,  28283,
 28289,  28297,  28307,  28309,  28319,  28349,  28351,  28387,  28393,  28403,
 28409,  28411,  28429,  28433,  28439,  28447,  28463,  28477,  28493,  28499,
 28513,  28517,  28537,  28541,  28547,  28549,  28559,  28571,  28573,  28579,
 28591,  28597,  28603,  28607,  28619,  28621,  28627,  28631,  28643,  28649,
 28657,  28661,  28663,  28669,  28687,  28697,  28703,  28711,  28723,  28729,
 28751,  28753,  28759,  28771,  28789,  28793,  28807,  28813,  28817,  28837,
 28843,  28859,  28867,  28871,  28879,  28901,  28909,  28921,  28927,  28933,
 28949,  28961,  28979,  29009,  29017,  29021,  29023,  29027,  29033,  29059,
 29063,  29077,  29101,  29123,  29129,  29131,  29137,  29147,  29153,  29167,
 29173,  29179,  29191,  29201,  29207,  29209,  29221,  29231,  29243,  29251,
 29269,  29287,  29297,  29303,  29311,  29327,  29333,  29339,  29347,  29363,
 29383,  29387,  29389,  29399,  29401,  29411,  29423,  29429,  29437,  29443,
 29453,  29473,  29483,  29501,  29527,  29531,  29537,  29567,  29569,  29573,
 29581,  29587,  29599,  29611,  29629,  29633,  29641,  29663,  29669,  29671,
 29683,  29717,  29723,  29741,  29753,  29759,  29761,  29789,  29803,  29819,
 29833,  29837,  29851,  29863,  29867,  29873,  29879,  29881,  29917,  29921,
 29927,  29947,  29959,  29983,  29989,  30011,  30013,  30029,  30047,  30059,
 30071,  30089,  30091,  30097,  30103,  30109,  30113,  30119,  30133,  30137,
 30139,  30161,  30169,  30181,  30187,  30197,  30203,  30211,  30223,  30241,
 30253,  30259,  30269,  30271,  30293,  30307,  30313,  30319,  30323,  30341,
 30347,  30367,  30389,  30391,  30403,  30427,  30431,  30449,  30467,  30469,
 30491,  30493,  30497,  30509,  30517,  30529,  30539,  30553,  30557,  30559,
 30577,  30593,  30631,  30637,  30643,  30649,  30661,  30671,  30677,  30689,
 30697,  30703,  30707,  30713,  30727,  30757,  30763,  30773,  30781,  30803,
 30809,  30817,  30829,  30839,  30841,  30851,  30853,  30859,  30869,  30871,
 30881,  30893,  30911,  30931,  30937,  30941,  30949,  30971,  30977,  30983,
 31013,  31019,  31033,  31039,  31051,  31063,  31069,  31079,  31081,  31091,
 31121,  31123,  31139,  31147,  31151,  31153,  31159,  31177,  31181,  31183,
 31189,  31193,  31219,  31223,  31231,  31237,  31247,  31249,  31253,  31259,
 31267,  31271,  31277,  31307,  31319,  31321,  31327,  31333,  31337,  31357,
 31379,  31387,  31391,  31393,  31397,  31469,  31477,  31481,  31489,  31511,
 31513,  31517,  31531,  31541,  31543,  31547,  31567,  31573,  31583,  31601,
 31607,  31627,  31643,  31649,  31657,  31663,  31667,  31687,  31699,  31721,
 31723,  31727,  31729,  31741,  31751,  31769,  31771,  31793,  31799,  31817,
 31847,  31849,  31859,  31873,  31883,  31891,  31907,  31957,  31963,  31973,
 31981,  31991,  32003,  32009,  32027,  32029,  32051,  32057,  32059,  32063,
 32069,  32077,  32083,  32089,  32099,  32117,  32119,  32141,  32143,  32159,
 32173,  32183,  32189,  32191,  32203,  32213,  32233,  32237,  32251,  32257,
 32261,  32297,  32299,  32303,  32309,  32321,  32323,  32327,  32341,  32353,
 32359,  32363,  32369,  32371,  32377,  32381,  32401,  32411,  32413,  32423,
 32429,  32441,  32443,  32467,  32479,  32491,  32497,  32503,  32507,  32531,
 32533,  32537,  32561,  32563,  32569,  32573,  32579,  32587,  32603,  32609,
 32611,  32621,  32633,  32647,  32653,  32687,  32693,  32707,  32713,  32717,
 32719,  32749,  32771,  32779,  32783,  32789,  32797,  32801,  32803,  32831,
 32833,  32839,  32843,  32869,  32887,  32909,  32911,  32917,  32933,  32939,
 32941,  32957,  32969,  32971,  32983,  32987,  32993,  32999,  33013,  33023,
 33029,  33037,  33049,  33053,  33071,  33073,  33083,  33091,  33107,  33113,
 33119,  33149,  33151,  33161,  33179,  33181,  33191,  33199,  33203,  33211,
 33223,  33247,  33287,  33289,  33301,  33311,  33317,  33329,  33331,  33343,
 33347,  33349,  33353,  33359,  33377,  33391,  33403,  33409,  33413,  33427,
 33457,  33461,  33469,  33479,  33487,  33493,  33503,  33521,  33529,  33533,
 33547,  33563,  33569,  33577,  33581,  33587,  33589,  33599,  33601,  33613,
 33617,  33619,  33623,  33629,  33637,  33641,  33647,  33679,  33703,  33713,
 33721,  33739,  33749,  33751,  33757,  33767,  33769,  33773,  33791,  33797,
 33809,  33811,  33827,  33829,  33851,  33857,  33863,  33871,  33889,  33893,
 33911,  33923,  33931,  33937,  33941,  33961,  33967,  33997,  34019,  34031,
 34033,  34039,  34057,  34061,  34123,  34127,  34129,  34141,  34147,  34157,
 34159,  34171,  34183,  34211,  34213,  34217,  34231,  34253,  34259,  34261,
 34267,  34273,  34283,  34297,  34301,  34303,  34313,  34319,  34327,  34337,
 34351,  34361,  34367,  34369,  34381,  34403,  34421,  34429,  34439,  34457,
 34469,  34471,  34483,  34487,  34499,  34501,  34511,  34513,  34519,  34537,
 34543,  34549,  34583,  34589,  34591,  34603,  34607,  34613,  34631,  34649,
 34651,  34667,  34673,  34679,  34687,  34693,  34703,  34721,  34729,  34739,
 34747,  34757,  34759,  34763,  34781,  34807,  34819,  34841,  34843,  34847,
 34849,  34871,  34877,  34883,  34897,  34913,  34919,  34939,  34949,  34961,
 34963,  34981,  35023,  35027,  35051,  35053,  35059,  35069,  35081,  35083,
 35089,  35099,  35107,  35111,  35117,  35129,  35141,  35149,  35153,  35159,
 35171,  35201,  35221,  35227,  35251,  35257,  35267,  35279,  35281,  35291,
 35311,  35317,  35323,  35327,  35339,  35353,  35363,  35381,  35393,  35401,
 35407,  35419,  35423,  35437,  35447,  35449,  35461,  35491,  35507,  35509,
 35521,  35527,  35531,  35533,  35537,  35543,  35569,  35573,  35591,  35593,
 35597,  35603,  35617,  35671,  35677,  35729,  35731,  35747,  35753,  35759,
 35771,  35797,  35801,  35803,  35809,  35831,  35837,  35839,  35851,  35863,
 35869,  35879,  35897,  35899,  35911,  35923,  35933,  35951,  35963,  35969,
 35977,  35983,  35993,  35999,  36007,  36011,  36013,  36017,  36037,  36061,
 36067,  36073,  36083,  36097,  36107,  36109,  36131,  36137,  36151,  36161,
 36187,  36191,  36209,  36217,  36229,  36241,  36251,  36263,  36269,  36277,
 36293,  36299,  36307,  36313,  36319,  36341,  36343,  36353,  36373,  36383,
 36389,  36433,  36451,  36457,  36467,  36469,  36473,  36479,  36493,  36497,
 36523,  36527,  36529,  36541,  36551,  36559,  36563,  36571,  36583,  36587,
 36599,  36607,  36629,  36637,  36643,  36653,  36671,  36677,  36683,  36691,
 36697,  36709,  36713,  36721,  36739,  36749,  36761,  36767,  36779,  36781,
 36787,  36791,  36793,  36809,  36821,  36833,  36847,  36857,  36871,  36877,
 36887,  36899,  36901,  36913,  36919,  36923,  36929,  36931,  36943,  36947,
 36973,  36979,  36997,  37003,  37013,  37019,  37021,  37039,  37049,  37057,
 37061,  37087,  37097,  37117,  37123,  37139,  37159,  37171,  37181,  37189,
 37199,  37201,  37217,  37223,  37243,  37253,  37273,  37277,  37307,  37309,
 37313,  37321,  37337,  37339,  37357,  37361,  37363,  37369,  37379,  37397,
 37409,  37423,  37441,  37447,  37463,  37483,  37489,  37493,  37501,  37507,
 37511,  37517,  37529,  37537,  37547,  37549,  37561,  37567,  37571,  37573,
 37579,  37589,  37591,  37607,  37619,  37633,  37643,  37649,  37657,  37663,
 37691,  37693,  37699,  37717,  37747,  37781,  37783,  37799,  37811,  37813,
 37831,  37847,  37853,  37861,  37871,  37879,  37889,  37897,  37907,  37951,
 37957,  37963,  37967,  37987,  37991,  37993,  37997,  38011,  38039,  38047,
 38053,  38069,  38083,  38113,  38119,  38149,  38153,  38167,  38177,  38183,
 38189,  38197,  38201,  38219,  38231,  38237,  38239,  38261,  38273,  38281,
 38287,  38299,  38303,  38317,  38321,  38327,  38329,  38333,  38351,  38371,
 38377,  38393,  38431,  38447,  38449,  38453,  38459,  38461,  38501,  38543,
 38557,  38561,  38567,  38569,  38593,  38603,  38609,  38611,  38629,  38639,
 38651,  38653,  38669,  38671,  38677,  38693,  38699,  38707,  38711,  38713,
 38723,  38729,  38737,  38747,  38749,  38767,  38783,  38791,  38803,  38821,
 38833,  38839,  38851,  38861,  38867,  38873,  38891,  38903,  38917,  38921,
 38923,  38933,  38953,  38959,  38971,  38977,  38993,  39019,  39023,  39041,
 39043,  39047,  39079,  39089,  39097,  39103,  39107,  39113,  39119,  39133,
 39139,  39157,  39161,  39163,  39181,  39191,  39199,  39209,  39217,  39227,
 39229,  39233,  39239,  39241,  39251,  39293,  39301,  39313,  39317,  39323,
 39341,  39343,  39359,  39367,  39371,  39373,  39383,  39397,  39409,  39419,
 39439,  39443,  39451,  39461,  39499,  39503,  39509,  39511,  39521,  39541,
 39551,  39563,  39569,  39581,  39607,  39619,  39623,  39631,  39659,  39667,
 39671,  39679,  39703,  39709,  39719,  39727,  39733,  39749,  39761,  39769,
 39779,  39791,  39799,  39821,  39827,  39829,  39839,  39841,  39847,  39857,
 39863,  39869,  39877,  39883,  39887,  39901,  39929,  39937,  39953,  39971,
 39979,  39983,  39989,  40009,  40013,  40031,  40037,  40039,  40063,  40087,
 40093,  40099,  40111,  40123,  40127,  40129,  40151,  40153,  40163,  40169,
 40177,  40189,  40193,  40213,  40231,  40237,  40241,  40253,  40277,  40283,
 40289,  40343,  40351,  40357,  40361,  40387,  40423,  40427,  40429,  40433,
 40459,  40471,  40483,  40487,  40493,  40499,  40507,  40519,  40529,  40531,
 40543,  40559,  40577,  40583,  40591,  40597,  40609,  40627,  40637,  40639,
 40693,  40697,  40699,  40709,  40739,  40751,  40759,  40763,  40771,  40787,
 40801,  40813,  40819,  40823,  40829,  40841,  40847,  40849,  40853,  40867,
 40879,  40883,  40897,  40903,  40927,  40933,  40939,  40949,  40961,  40973,
 40993,  41011,  41017,  41023,  41039,  41047,  41051,  41057,  41077,  41081,
 41113,  41117,  41131,  41141,  41143,  41149,  41161,  41177,  41179,  41183,
 41189,  41201,  41203,  41213,  41221,  41227,  41231,  41233,  41243,  41257,
 41263,  41269,  41281,  41299,  41333,  41341,  41351,  41357,  41381,  41387,
 41389,  41399,  41411,  41413,  41443,  41453,  41467,  41479,  41491,  41507,
 41513,  41519,  41521,  41539,  41543,  41549,  41579,  41593,  41597,  41603,
 41609,  41611,  41617,  41621,  41627,  41641,  41647,  41651,  41659,  41669,
 41681,  41687,  41719,  41729,  41737,  41759,  41761,  41771,  41777,  41801,
 41809,  41813,  41843,  41849,  41851,  41863,  41879,  41887,  41893,  41897,
 41903,  41911,  41927,  41941,  41947,  41953,  41957,  41959,  41969,  41981,
 41983,  41999,  42013,  42017,  42019,  42023,  42043,  42061,  42071,  42073,
 42083,  42089,  42101,  42131,  42139,  42157,  42169,  42179,  42181,  42187,
 42193,  42197,  42209,  42221,  42223,  42227,  42239,  42257,  42281,  42283,
 42293,  42299,  42307,  42323,  42331,  42337,  42349,  42359,  42373,  42379,
 42391,  42397,  42403,  42407,  42409,  42433,  42437,  42443,  42451,  42457,
 42461,  42463,  42467,  42473,  42487,  42491,  42499,  42509,  42533,  42557,
 42569,  42571,  42577,  42589,  42611,  42641,  42643,  42649,  42667,  42677,
 42683,  42689,  42697,  42701,  42703,  42709,  42719,  42727,  42737,  42743,
 42751,  42767,  42773,  42787,  42793,  42797,  42821,  42829,  42839,  42841,
 42853,  42859,  42863,  42899,  42901,  42923,  42929,  42937,  42943,  42953,
 42961,  42967,  42979,  42989,  43003,  43013,  43019,  43037,  43049,  43051,
 43063,  43067,  43093,  43103,  43117,  43133,  43151,  43159,  43177,  43189,
 43201,  43207,  43223,  43237,  43261,  43271,  43283,  43291,  43313,  43319,
 43321,  43331,  43391,  43397,  43399,  43403,  43411,  43427,  43441,  43451,
 43457,  43481,  43487,  43499,  43517,  43541,  43543,  43573,  43577,  43579,
 43591,  43597,  43607,  43609,  43613,  43627,  43633,  43649,  43651,  43661,
 43669,  43691,  43711,  43717,  43721,  43753,  43759,  43777,  43781,  43783,
 43787,  43789,  43793,  43801,  43853,  43867,  43889,  43891,  43913,  43933,
 43943,  43951,  43961,  43963,  43969,  43973,  43987,  43991,  43997,  44017,
 44021,  44027,  44029,  44041,  44053,  44059,  44071,  44087,  44089,  44101,
 44111,  44119,  44123,  44129,  44131,  44159,  44171,  44179,  44189,  44201,
 44203,  44207,  44221,  44249,  44257,  44263,  44267,  44269,  44273,  44279,
 44281,  44293,  44351,  44357,  44371,  44381,  44383,  44389,  44417,  44449,
 44453,  44483,  44491,  44497,  44501,  44507,  44519,  44531,  44533,  44537,
 44543,  44549,  44563,  44579,  44587,  44617,  44621,  44623,  44633,  44641,
 44647,  44651,  44657,  44683,  44687,  44699,  44701,  44711,  44729,  44741,
 44753,  44771,  44773,  44777,  44789,  44797,  44809,  44819,  44839,  44843,
 44851,  44867,  44879,  44887,  44893,  44909,  44917,  44927,  44939,  44953,
 44959,  44963,  44971,  44983,  44987,  45007,  45013,  45053,  45061,  45077,
 45083,  45119,  45121,  45127,  45131,  45137,  45139,  45161,  45179,  45181,
 45191,  45197,  45233,  45247,  45259,  45263,  45281,  45289,  45293,  45307,
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 49667,  49669,  49681,  49697,  49711,  49727,  49739,  49741,  49747,  49757,
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 49871,  49877,  49891,  49919,  49921,  49927,  49937,  49939,  49943,  49957,
 49991,  49993,  49999,  50021,  50023,  50033,  50047,  50051,  50053,  50069,
 50077,  50087,  50093,  50101,  50111,  50119,  50123,  50129,  50131,  50147,
 50153,  50159,  50177,  50207,  50221,  50227,  50231,  50261,  50263,  50273,
 50287,  50291,  50311,  50321,  50329,  50333,  50341,  50359,  50363,  50377,
 50383,  50387,  50411,  50417,  50423,  50441,  50459,  50461,  50497,  50503,
 50513,  50527,  50539,  50543,  50549,  50551,  50581,  50587,  50591,  50593,
 50599,  50627,  50647,  50651,  50671,  50683,  50707,  50723,  50741,  50753,
 50767,  50773,  50777,  50789,  50821,  50833,  50839,  50849,  50857,  50867,
 50873,  50891,  50893,  50909,  50923,  50929,  50951,  50957,  50969,  50971,
 50989,  50993,  51001,  51031,  51043,  51047,  51059,  51061,  51071,  51109,
 51131,  51133,  51137,  51151,  51157,  51169,  51193,  51197,  51199,  51203,
 51217,  51229,  51239,  51241,  51257,  51263,  51283,  51287,  51307,  51329,
 51341,  51343,  51347,  51349,  51361,  51383,  51407,  51413,  51419,  51421,
 51427,  51431,  51437,  51439,  51449,  51461,  51473,  51479,  51481,  51487,
 51503,  51511,  51517,  51521,  51539,  51551,  51563,  51577,  51581,  51593,
 51599,  51607,  51613,  51631,  51637,  51647,  51659,  51673,  51679,  51683,
 51691,  51713,  51719,  51721,  51749,  51767,  51769,  51787,  51797,  51803,
 51817,  51827,  51829,  51839,  51853,  51859,  51869,  51871,  51893,  51899,
 51907,  51913,  51929,  51941,  51949,  51971,  51973,  51977,  51991,  52009,
 52021,  52027,  52051,  52057,  52067,  52069,  52081,  52103,  52121,  52127,
 52147,  52153,  52163,  52177,  52181,  52183,  52189,  52201,  52223,  52237,
 52249,  52253,  52259,  52267,  52289,  52291,  52301,  52313,  52321,  52361,
 52363,  52369,  52379,  52387,  52391,  52433,  52453,  52457,  52489,  52501,
 52511,  52517,  52529,  52541,  52543,  52553,  52561,  52567,  52571,  52579,
 52583,  52609,  52627,  52631,  52639,  52667,  52673,  52691,  52697,  52709,
 52711,  52721,  52727,  52733,  52747,  52757,  52769,  52783,  52807,  52813,
 52817,  52837,  52859,  52861,  52879,  52883,  52889,  52901,  52903,  52919,
 52937,  52951,  52957,  52963,  52967,  52973,  52981,  52999,  53003,  53017,
 53047,  53051,  53069,  53077,  53087,  53089,  53093,  53101,  53113,  53117,
 53129,  53147,  53149,  53161,  53171,  53173,  53189,  53197,  53201,  53231,
 53233,  53239,  53267,  53269,  53279,  53281,  53299,  53309,  53323,  53327,
 53353,  53359,  53377,  53381,  53401,  53407,  53411,  53419,  53437,  53441,
 53453,  53479,  53503,  53507,  53527,  53549,  53551,  53569,  53591,  53593,
 53597,  53609,  53611,  53617,  53623,  53629,  53633,  53639,  53653,  53657,
 53681,  53693,  53699,  53717,  53719,  53731,  53759,  53773,  53777,  53783,
 53791,  53813,  53819,  53831,  53849,  53857,  53861,  53881,  53887,  53891,
 53897,  53899,  53917,  53923,  53927,  53939,  53951,  53959,  53987,  53993,
 54001,  54011,  54013,  54037,  54049,  54059,  54083,  54091,  54101,  54121,
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 54449,  54469,  54493,  54497,  54499,  54503,  54517,  54521,  54539,  54541,
 54547,  54559,  54563,  54577,  54581,  54583,  54601,  54617,  54623,  54629,
 54631,  54647,  54667,  54673,  54679,  54709,  54713,  54721,  54727,  54751,
 54767,  54773,  54779,  54787,  54799,  54829,  54833,  54851,  54869,  54877,
 54881,  54907,  54917,  54919,  54941,  54949,  54959,  54973,  54979,  54983,
 55001,  55009,  55021,  55049,  55051,  55057,  55061,  55073,  55079,  55103,
 55109,  55117,  55127,  55147,  55163,  55171,  55201,  55207,  55213,  55217,
 55219,  55229,  55243,  55249,  55259,  55291,  55313,  55331,  55333,  55337,
 55339,  55343,  55351,  55373,  55381,  55399,  55411,  55439,  55441,  55457,
 55469,  55487,  55501,  55511,  55529,  55541,  55547,  55579,  55589,  55603,
 55609,  55619,  55621,  55631,  55633,  55639,  55661,  55663,  55667,  55673,
 55681,  55691,  55697,  55711,  55717,  55721,  55733,  55763,  55787,  55793,
 55799,  55807,  55813,  55817,  55819,  55823,  55829,  55837,  55843,  55849,
 55871,  55889,  55897,  55901,  55903,  55921,  55927,  55931,  55933,  55949,
 55967,  55987,  55997,  56003,  56009,  56039,  56041,  56053,  56081,  56087,
 56093,  56099,  56101,  56113,  56123,  56131,  56149,  56167,  56171,  56179,
 56197,  56207,  56209,  56237,  56239,  56249,  56263,  56267,  56269,  56299,
 56311,  56333,  56359,  56369,  56377,  56383,  56393,  56401,  56417,  56431,
 56437,  56443,  56453,  56467,  56473,  56477,  56479,  56489,  56501,  56503,
 56509,  56519,  56527,  56531,  56533,  56543,  56569,  56591,  56597,  56599,
 56611,  56629,  56633,  56659,  56663,  56671,  56681,  56687,  56701,  56711,
 56713,  56731,  56737,  56747,  56767,  56773,  56779,  56783,  56807,  56809,
 56813,  56821,  56827,  56843,  56857,  56873,  56891,  56893,  56897,  56909,
 56911,  56921,  56923,  56929,  56941,  56951,  56957,  56963,  56983,  56989,
 56993,  56999,  57037,  57041,  57047,  57059,  57073,  57077,  57089,  57097,
 57107,  57119,  57131,  57139,  57143,  57149,  57163,  57173,  57179,  57191,
 57193,  57203,  57221,  57223,  57241,  57251,  57259,  57269,  57271,  57283,
 57287,  57301,  57329,  57331,  57347,  57349,  57367,  57373,  57383,  57389,
 57397,  57413,  57427,  57457,  57467,  57487,  57493,  57503,  57527,  57529,
 57557,  57559,  57571,  57587,  57593,  57601,  57637,  57641,  57649,  57653,
 57667,  57679,  57689,  57697,  57709,  57713,  57719,  57727,  57731,  57737,
 57751,  57773,  57781,  57787,  57791,  57793,  57803,  57809,  57829,  57839,
 57847,  57853,  57859,  57881,  57899,  57901,  57917,  57923,  57943,  57947,
 57973,  57977,  57991,  58013,  58027,  58031,  58043,  58049,  58057,  58061,
 58067,  58073,  58099,  58109,  58111,  58129,  58147,  58151,  58153,  58169,
 58171,  58189,  58193,  58199,  58207,  58211,  58217,  58229,  58231,  58237,
 58243,  58271,  58309,  58313,  58321,  58337,  58363,  58367,  58369,  58379,
 58391,  58393,  58403,  58411,  58417,  58427,  58439,  58441,  58451,  58453,
 58477,  58481,  58511,  58537,  58543,  58549,  58567,  58573,  58579,  58601,
 58603,  58613,  58631,  58657,  58661,  58679,  58687,  58693,  58699,  58711,
 58727,  58733,  58741,  58757,  58763,  58771,  58787,  58789,  58831,  58889,
 58897,  58901,  58907,  58909,  58913,  58921,  58937,  58943,  58963,  58967,
 58979,  58991,  58997,  59009,  59011,  59021,  59023,  59029,  59051,  59053,
 59063,  59069,  59077,  59083,  59093,  59107,  59113,  59119,  59123,  59141,
 59149,  59159,  59167,  59183,  59197,  59207,  59209,  59219,  59221,  59233,
 59239,  59243,  59263,  59273,  59281,  59333,  59341,  59351,  59357,  59359,
 59369,  59377,  59387,  59393,  59399,  59407,  59417,  59419,  59441,  59443,
 59447,  59453,  59467,  59471,  59473,  59497,  59509,  59513,  59539,  59557,
 59561,  59567,  59581,  59611,  59617,  59621,  59627,  59629,  59651,  59659,
 59663,  59669,  59671,  59693,  59699,  59707,  59723,  59729,  59743,  59747,
 59753,  59771,  59779,  59791,  59797,  59809,  59833,  59863,  59879,  59887,
 59921,  59929,  59951,  59957,  59971,  59981,  59999,  60013,  60017,  60029,
 60037,  60041,  60077,  60083,  60089,  60091,  60101,  60103,  60107,  60127,
 60133,  60139,  60149,  60161,  60167,  60169,  60209,  60217,  60223,  60251,
 60257,  60259,  60271,  60289,  60293,  60317,  60331,  60337,  60343,  60353,
 60373,  60383,  60397,  60413,  60427,  60443,  60449,  60457,  60493,  60497,
 60509,  60521,  60527,  60539,  60589,  60601,  60607,  60611,  60617,  60623,
 60631,  60637,  60647,  60649,  60659,  60661,  60679,  60689,  60703,  60719,
 60727,  60733,  60737,  60757,  60761,  60763,  60773,  60779,  60793,  60811,
 60821,  60859,  60869,  60887,  60889,  60899,  60901,  60913,  60917,  60919,
 60923,  60937,  60943,  60953,  60961,  61001,  61007,  61027,  61031,  61043,
 61051,  61057,  61091,  61099,  61121,  61129,  61141,  61151,  61153,  61169,
 61211,  61223,  61231,  61253,  61261,  61283,  61291,  61297,  61331,  61333,
 61339,  61343,  61357,  61363,  61379,  61381,  61403,  61409,  61417,  61441,
 61463,  61469,  61471,  61483,  61487,  61493,  61507,  61511,  61519,  61543,
 61547,  61553,  61559,  61561,  61583,  61603,  61609,  61613,  61627,  61631,
 61637,  61643,  61651,  61657,  61667,  61673,  61681,  61687,  61703,  61717,
 61723,  61729,  61751,  61757,  61781,  61813,  61819,  61837,  61843,  61861,
 61871,  61879,  61909,  61927,  61933,  61949,  61961,  61967,  61979,  61981,
 61987,  61991,  62003,  62011,  62017,  62039,  62047,  62053,  62057,  62071,
 62081,  62099,  62119,  62129,  62131,  62137,  62141,  62143,  62171,  62189,
 62191,  62201,  62207,  62213,  62219,  62233,  62273,  62297,  62299,  62303,
 62311,  62323,  62327,  62347,  62351,  62383,  62401,  62417,  62423,  62459,
 62467,  62473,  62477,  62483,  62497,  62501,  62507,  62533,  62539,  62549,
 62563,  62581,  62591,  62597,  62603,  62617,  62627,  62633,  62639,  62653,
 62659,  62683,  62687,  62701,  62723,  62731,  62743,  62753,  62761,  62773,
 62791,  62801,  62819,  62827,  62851,  62861,  62869,  62873,  62897,  62903,
 62921,  62927,  62929,  62939,  62969,  62971,  62981,  62983,  62987,  62989,
 63029,  63031,  63059,  63067,  63073,  63079,  63097,  63103,  63113,  63127,
 63131,  63149,  63179,  63197,  63199,  63211,  63241,  63247,  63277,  63281,
 63299,  63311,  63313,  63317,  63331,  63337,  63347,  63353,  63361,  63367,
 63377,  63389,  63391,  63397,  63409,  63419,  63421,  63439,  63443,  63463,
 63467,  63473,  63487,  63493,  63499,  63521,  63527,  63533,  63541,  63559,
 63577,  63587,  63589,  63599,  63601,  63607,  63611,  63617,  63629,  63647,
 63649,  63659,  63667,  63671,  63689,  63691,  63697,  63703,  63709,  63719,
 63727,  63737,  63743,  63761,  63773,  63781,  63793,  63799,  63803,  63809,
 63823,  63839,  63841,  63853,  63857,  63863,  63901,  63907,  63913,  63929,
 63949,  63977,  63997,  64007,  64013,  64019,  64033,  64037,  64063,  64067,
 64081,  64091,  64109,  64123,  64151,  64153,  64157,  64171,  64187,  64189,
 64217,  64223,  64231,  64237,  64271,  64279,  64283,  64301,  64303,  64319,
 64327,  64333,  64373,  64381,  64399,  64403,  64433,  64439,  64451,  64453,
 64483,  64489,  64499,  64513,  64553,  64567,  64577,  64579,  64591,  64601,
 64609,  64613,  64621,  64627,  64633,  64661,  64663,  64667,  64679,  64693,
 64709,  64717,  64747,  64763,  64781,  64783,  64793,  64811,  64817,  64849,
 64853,  64871,  64877,  64879,  64891,  64901,  64919,  64921,  64927,  64937,
 64951,  64969,  64997,  65003,  65011,  65027,  65029,  65033,  65053,  65063,
 65071,  65089,  65099,  65101,  65111,  65119,  65123,  65129,  65141,  65147,
 65167,  65171,  65173,  65179,  65183,  65203,  65213,  65239,  65257,  65267,
 65269,  65287,  65293,  65309,  65323,  65327,  65353,  65357,  65371,  65381,
 65393,  65407,  65413,  65419,  65423,  65437,  65447,  65449,  65479,  65497,
 65519,  65521,  65537,  65539,  65543,  65551,  65557,  65563,  65579,  65581,
 65587,  65599,  65609,  65617,  65629,  65633,  65647,  65651,  65657,  65677,
 65687,  65699,  65701,  65707,  65713,  65717,  65719,  65729,  65731,  65761,
 65777,  65789,  65809,  65827,  65831,  65837,  65839,  65843,  65851,  65867,
 65881,  65899,  65921,  65927,  65929,  65951,  65957,  65963,  65981,  65983,
 65993,  66029,  66037,  66041,  66047,  66067,  66071,  66083,  66089,  66103,
 66107,  66109,  66137,  66161,  66169,  66173,  66179,  66191,  66221,  66239,
 66271,  66293,  66301,  66337,  66343,  66347,  66359,  66361,  66373,  66377,
 66383,  66403,  66413,  66431,  66449,  66457,  66463,  66467,  66491,  66499,
 66509,  66523,  66529,  66533,  66541,  66553,  66569,  66571,  66587,  66593,
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 88883,  88897,  88903,  88919,  88937,  88951,  88969,  88993,  88997,  89003,
 89009,  89017,  89021,  89041,  89051,  89057,  89069,  89071,  89083,  89087,
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 89867,  89891,  89897,  89899,  89909,  89917,  89923,  89939,  89959,  89963,
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 90499,  90511,  90523,  90527,  90529,  90533,  90547,  90583,  90599,  90617,
 90619,  90631,  90641,  90647,  90659,  90677,  90679,  90697,  90703,  90709,
 90731,  90749,  90787,  90793,  90803,  90821,  90823,  90833,  90841,  90847,
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 91811,  91813,  91823,  91837,  91841,  91867,  91873,  91909,  91921,  91939,
 91943,  91951,  91957,  91961,  91967,  91969,  91997,  92003,  92009,  92033,
 92041,  92051,  92077,  92083,  92107,  92111,  92119,  92143,  92153,  92173,
 92177,  92179,  92189,  92203,  92219,  92221,  92227,  92233,  92237,  92243,
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 92459,  92461,  92467,  92479,  92489,  92503,  92507,  92551,  92557,  92567,
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100129, 100151, 100153, 100169, 100183, 100189, 100193, 100207, 100213, 100237,
100267, 100271, 100279, 100291, 100297, 100313, 100333, 100343, 100357, 100361,
100363, 100379, 100391, 100393, 100403, 100411, 100417, 100447, 100459, 100469,
100483, 100493, 100501, 100511, 100517, 100519, 100523, 100537, 100547, 100549,
100559, 100591, 100609, 100613, 100621, 100649, 100669, 100673, 100693, 100699,
100703, 100733, 100741, 100747, 100769, 100787, 100799, 100801, 100811, 100823,
100829, 100847, 100853, 100907, 100913, 100927, 100931, 100937, 100943, 100957,
100981, 100987, 100999, 101009, 101021, 101027, 101051, 101063, 101081, 101089,
101107, 101111, 101113, 101117, 101119, 101141, 101149, 101159, 101161, 101173,
101183, 101197, 101203, 101207, 101209, 101221, 101267, 101273, 101279, 101281,
101287, 101293, 101323, 101333, 101341, 101347, 101359, 101363, 101377, 101383,
101399, 101411, 101419, 101429, 101449, 101467, 101477, 101483, 101489, 101501,
101503, 101513, 101527, 101531, 101533, 101537, 101561, 101573, 101581, 101599,
101603, 101611, 101627, 101641, 101653, 101663, 101681, 101693, 101701, 101719,
101723, 101737, 101741, 101747, 101749, 101771, 101789, 101797, 101807, 101833,
101837, 101839, 101863, 101869, 101873, 101879, 101891, 101917, 101921, 101929,
101939, 101957, 101963, 101977, 101987, 101999, 102001, 102013, 102019, 102023,
102031, 102043, 102059, 102061, 102071, 102077, 102079, 102101, 102103, 102107,
102121, 102139, 102149, 102161, 102181, 102191, 102197, 102199, 102203, 102217,
102229, 102233, 102241, 102251, 102253, 102259, 102293, 102299, 102301, 102317,
102329, 102337, 102359, 102367, 102397, 102407, 102409, 102433, 102437, 102451,
102461, 102481, 102497, 102499, 102503, 102523, 102533, 102539, 102547, 102551,
102559, 102563, 102587, 102593, 102607, 102611, 102643, 102647, 102653, 102667,
102673, 102677, 102679, 102701, 102761, 102763, 102769, 102793, 102797, 102811,
102829, 102841, 102859, 102871, 102877, 102881, 102911, 102913, 102929, 102931,
102953, 102967, 102983, 103001, 103007, 103043, 103049, 103067, 103069, 103079,
103087, 103091, 103093, 103099, 103123, 103141, 103171, 103177, 103183, 103217,
103231, 103237, 103289, 103291, 103307, 103319, 103333, 103349, 103357, 103387,
103391, 103393, 103399, 103409, 103421, 103423, 103451, 103457, 103471, 103483,
103511, 103529, 103549, 103553, 103561, 103567, 103573, 103577, 103583, 103591,
103613, 103619, 103643, 103651, 103657, 103669, 103681, 103687, 103699, 103703,
103723, 103769, 103787, 103801, 103811, 103813, 103837, 103841, 103843, 103867,
103889, 103903, 103913, 103919, 103951, 103963, 103967, 103969, 103979, 103981,
103991, 103993, 103997, 104003, 104009, 104021, 104033, 104047, 104053, 104059,
104087, 104089, 104107, 104113, 104119, 104123, 104147, 104149, 104161, 104173,
104179, 104183, 104207, 104231, 104233, 104239, 104243, 104281, 104287, 104297,
104309, 104311, 104323, 104327, 104347, 104369, 104381, 104383, 104393, 104399,
104417, 104459, 104471, 104473, 104479, 104491, 104513, 104527, 104537, 104543,
104549, 104551, 104561, 104579, 104593, 104597, 104623, 104639, 104651, 104659,
104677, 104681, 104683, 104693, 104701, 104707, 104711, 104717, 104723, 104729,
)