def nth_prime(n):
assert n > 0
bound = max([1, int(math.log(n) * n)])
primes = sieve(bound)
while len(primes) < n:
bound *= 2
primes = sieve(bound)
return primes[n - 1]
def test3():
s = sieve.sieve()
i = iter(s)
for x in sieve.sieve():
if x > 151:
break
assert x == 157
print x
def test2():
s = sieve.sieve()
i = iter(s)
for x in sieve.sieve():
if x > 10:
break
assert x == 11
print x
def agen(i, k=3):
primes = sieve(k)
if k not in sieve(k + 1):
return
while True:
yield int(i)
if i <= 1:
break
for p in primes:
if 0 == i % p:
i = i / p
break
else:
i = k * i + 1
def factorize(p):
"""Returns a list of the prime factors of p.
If a number is a factor multiple times (such as with squares or cubes),
it will be included in the factor list once for each multiplication.
If p is itself prime, returns a list consisting of one element, p.
Args:
p: integer, the number for which to find prime factors.
Returns:
A list of the prime factors for p, or an empty list if p < 2.
Raises:
ValueError: p is < 2.
"""
primes = sieve(p)
# If p is prime, it should be at the end of the list and will have
# no other factors
if primes[-1] == p:
return [p]
factors = []
for prime in primes:
if p == 1:
break
# Use while instead of if for squares, cubes, etc
while p % prime == 0:
factors.append(prime)
p /= prime
return factors
def test1():
s = sieve.sieve()
i = iter(s)
n = i.next()
print n
assert n == 3
print "***Test 1 passed!***"
def test3():
s = sieve.sieve()
i = iter(s)
for x in range(8):
i.next()
assert i.next() == 29
def test1():
s = sieve.sieve() #make a sieve object from file sieve
i = iter(s) #make iterator
val=i.next() #run iterator once, put value in val
assert val == 3 #check if value is 3. if not 3, fail.
print "Value should be 3, actual value is:"
print val
def test3():
sieveInstance = sieve.sieve()
i = iter(sieveInstance)
i.next()
i.next()
assert i.next() == 7
print "Test 3 Passes"
def primes2(pf, n):
if pf <= 0:
raise ValueError
elif n <= 2**pf:
return []
P = sieve(n)
if pf == 1:
return P
powP = [P]
func = lambda p: (lambda x: (powP[p][x] * P[x] < n))
for p in range(pf):
tmpP = binary_takewhile(powP[p], func(p))
powP.append(tmpP * P[:len(tmpP)])
del tmpP
T = np.zeros(n, dtype=np.uint8)
for Ps in powP:
for p in Ps:
T[p::p] += 1
del powP
return np.where(T == pf)[0].astype(np.uint64)
def test3():
"3 is a bigger number than 2 or 1, and thats a fact"
s = sieve.sieve()
i = iter(s)
i.next()
i.next()
assert i.next() == 7
def test1():
s = sieve.sieve() #create an object of type sieve
i = iter(s) #create an interator for the object sieve
#Check if next value is equal to 3
#Error would occur if the value is not equal to 3
assert i.next() == 3
def test2():
s = sieve.sieve()
i = iter(s)
for x in range(4):
i.next()
assert i.next() == 13
def createAlgsList(self):
# First we populate the list of algorithms with those created
# extending GeoAlgorithm directly (those that execute GDAL
# using the console)
self.preloadedAlgs = [nearblack(), information(), warp(), translate(),
rgb2pct(), pct2rgb(), merge(), buildvrt(), polygonize(), gdaladdo(),
ClipByExtent(), ClipByMask(), contour(), rasterize(), proximity(),
sieve(), fillnodata(), ExtractProjection(), gdal2xyz(),
hillshade(), slope(), aspect(), tri(), tpi(), roughness(),
ColorRelief(), GridInvDist(), GridAverage(), GridNearest(),
GridDataMetrics(), gdaltindex(), gdalcalc(), rasterize_over(),
# ----- OGR tools -----
OgrInfo(), Ogr2Ogr(), Ogr2OgrClip(), Ogr2OgrClipExtent(),
Ogr2OgrToPostGis(), Ogr2OgrToPostGisList(), Ogr2OgrPointsOnLines(),
Ogr2OgrBuffer(), Ogr2OgrDissolve(), Ogr2OgrOneSideBuffer(),
Ogr2OgrTableToPostGisList(), OgrSql(),
]
# And then we add those that are created as python scripts
folder = self.scriptsFolder()
if os.path.exists(folder):
for descriptionFile in os.listdir(folder):
if descriptionFile.endswith('py'):
try:
fullpath = os.path.join(self.scriptsFolder(),
descriptionFile)
alg = GdalScriptAlgorithm(fullpath)
self.preloadedAlgs.append(alg)
except WrongScriptException as e:
ProcessingLog.addToLog(ProcessingLog.LOG_ERROR, e.msg)
def test2():
s = sieve.sieve()
i = iter(s)
print i.next()
print i.next()
assert i.next() == 7
print "***Test 2 passed!***"
def test(r):
print 'Running test for', r, 'items'
for n, i in zip(range(r), sieve.sieve()):
print i
print '\n'
def createAlgsList(self):
# First we populate the list of algorithms with those created
# extending GeoAlgorithm directly (those that execute GDAL
# using the console)
self.preloadedAlgs = [nearblack(), information(), warp(), translate(),
rgb2pct(), pct2rgb(), merge(), polygonize(), gdaladdo(),
ClipByExtent(), ClipByMask(), contour(), rasterize(), proximity(),
sieve(), fillnodata(), ExtractProjection(), gdal2xyz(),
hillshade(), slope(), aspect(), tri(), tpi(), roughness(),
ColorRelief(), GridInvDist(), GridAverage(), GridNearest(),
GridDataMetrics(),
# ----- OGR tools -----
OgrInfo(), Ogr2Ogr(), OgrSql(),
]
# And then we add those that are created as python scripts
folder = self.scriptsFolder()
if os.path.exists(folder):
for descriptionFile in os.listdir(folder):
if descriptionFile.endswith('py'):
try:
fullpath = os.path.join(self.scriptsFolder(),
descriptionFile)
alg = GdalScriptAlgorithm(fullpath)
self.preloadedAlgs.append(alg)
except WrongScriptException, e:
ProcessingLog.addToLog(ProcessingLog.LOG_ERROR, e.msg)
def primes4(pf, n):
if pf <= 0:
raise ValueError
elif n <= 2**pf:
return []
P = sieve(-(-n >> (pf - 1)))
if pf == 1:
return P
PH = [0] * (pf - 1)
sup = pow(n, 1 / pf)
pwpf = 2**pf
T = np.zeros(n - pwpf, dtype=np.bool)
func = lambda x: (Pprod * P[x + index] < n)
while P[PH[0]] < sup:
index = PH[-1]
Pprod = np.prod(P[PH])
if Pprod * P[index] < n:
T[(binary_takewhile(P[index:], func) * Pprod) - pwpf] = True
PH[-1] += 1
else:
for i in range(-2, -pf - 1, -1):
if PH[i] != index:
PH[i:] = [PH[i] + 1] * (-i)
break
del P
return np.add(np.nonzero(T)[0].astype(np.uint64), np.uint64(2**pf))
def test3():
"This one is impossibly difficult to understand"
# psych
s = sieve.sieve()
i = iter(s)
i.next()
i.next()
assert i.next() == 7
def test_sieve_200(self):
expected = [
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
]
actual = sieve(200)
self.assertEqual(expected, actual)
def test2():
s = sieve.sieve()
i = iter(s)
for x in range(3):
i.next()
assert i.next() == 11 # the fifth prime number should be 11
def test3():
s = sieve.sieve() # make a sieve object from file sieve
i = iter(s) # make iterator
val = i.next() # run iterator once, put value in val
assert val != 2 # check if value not 2
print "Value should not be 2, actual value is:"
print val
def test3():
primes_list = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
ctr = 1
for n, i in zip(sieve.sieve(), primes_list):
if ctr == 11:
break
assert n == i
ctr += 1
def prime_factors(n):
candidates = sieve(n + 1)
factors = []
for c in candidates:
while n % c == 0:
factors.append(c)
n /= c
return factors
def test2():
sve = sieve.sieve()
i = iter(sve)
nxt = i.next()
print nxt
nxt = i.next()
print nxt
assert nxt == 5
def test2():
s = sieve.sieve() # create an object of type sieve
s.next() # move to the next prime number (3)
# Check if next value is equal to 5
# Error would occur if the value is not equal to 5
assert s.next() == 5
def test2():
s = sieve.sieve() #create an object of type sieve
i = iter(s) #create an interator for the object sieve
i.next() #move to the next value of iterator 3
#Check if next value is equal to 5
#Error would occur if the value is not equal to 5
assert i.next() == 5
def test3(): s = sieve.sieve() i = iter(s) for n in range(13): i.next() assert i.next() == 47
def test2(): s = sieve.sieve() i = iter(s) for n in range(10): i.next() assert i.next() == 37
def test3():
s = sieve.sieve()
i = iter(s)
for c in range(10):
print i.next()
assert i.next() == 37
print "***Test 3 passed!***"
def prime_factors(n):
primes = sieve(math.ceil(math.sqrt(n)))
result = []
for p in primes:
while n % p == 0:
result.append(p)
n /= p
if n > 1:
result.append(n)
return result
def prime_factors(n):
'''Find the prime factors of N'''
factors = []
while n > 1:
for p in sieve(n):
if n % p == 0 or n == p:
factors.append(p)
n /= p
break
return factors
def createAlgsList(self):
# First we populate the list of algorithms with those created
# extending GeoAlgorithm directly (those that execute GDAL
# using the console)
self.preloadedAlgs = [
nearblack(),
information(),
warp(),
translate(),
rgb2pct(),
pct2rgb(),
merge(),
buildvrt(),
polygonize(),
gdaladdo(),
ClipByExtent(),
ClipByMask(),
contour(),
rasterize(),
proximity(),
sieve(),
fillnodata(),
ExtractProjection(),
gdal2xyz(),
hillshade(),
slope(),
aspect(),
tri(),
tpi(),
roughness(),
ColorRelief(),
GridInvDist(),
GridAverage(),
GridNearest(),
GridDataMetrics(),
gdaltindex(),
gdalcalc(),
rasterize_over(),
retile(),
gdal2tiles(),
# ----- OGR tools -----
OgrInfo(),
Ogr2Ogr(),
Ogr2OgrClip(),
Ogr2OgrClipExtent(),
Ogr2OgrToPostGis(),
Ogr2OgrToPostGisList(),
Ogr2OgrPointsOnLines(),
Ogr2OgrBuffer(),
Ogr2OgrDissolve(),
Ogr2OgrOneSideBuffer(),
Ogr2OgrTableToPostGisList(),
OgrSql(),
]
def main(n=1000000):
sieve_list=sieve(n)
count=11
primes=[]
start=10
while count!=0 or start>n:
if isTruncatableprime(start,sieve_list):
primes.append(start)
count-=1
start+=1
return primes
def createAlgsList(self):
# First we populate the list of algorithms with those created
# extending GeoAlgorithm directly (those that execute GDAL
# using the console)
self.preloadedAlgs = [
nearblack(),
information(),
warp(),
translate(),
rgb2pct(),
pct2rgb(),
merge(),
buildvrt(),
polygonize(),
gdaladdo(),
ClipByExtent(),
ClipByMask(),
contour(),
rasterize(),
proximity(),
sieve(),
fillnodata(),
ExtractProjection(),
gdal2xyz(),
hillshade(),
slope(),
aspect(),
tri(),
tpi(),
roughness(),
ColorRelief(),
GridInvDist(),
GridAverage(),
GridNearest(),
GridDataMetrics(),
gdaltindex(),
gdalcalc(),
rasterize_over(),
retile(),
gdal2tiles(),
# ----- OGR tools -----
OgrInfo(),
Ogr2Ogr(),
Ogr2OgrClip(),
Ogr2OgrClipExtent(),
Ogr2OgrToPostGis(),
Ogr2OgrToPostGisList(),
Ogr2OgrPointsOnLines(),
Ogr2OgrBuffer(),
Ogr2OgrDissolve(),
Ogr2OgrOneSideBuffer(),
Ogr2OgrTableToPostGisList(),
OgrSql(),
]
def main(n):
oddities=[]
prime_sieve=sieve(n)
odd_sieve=gen_odd_composites(n,prime_sieve)
for odd in odd_sieve:
count=0
for prime in suitable_primes(odd ,prime_sieve):
if isPerfectSquare((odd-prime)/2):
count+=1
if count==0:
oddities.append(odd)
return oddities
def p_69():
max_n_phi = 0
n_max = 0
M = 10**6
cached_primes = sieve(M + 1)
for n in xrange(2, M):
n_phi = float(n)/totient(n, cached_primes)
if n_phi > max_n_phi:
max_n_phi = n_phi
n_max = n
print n_max, max_n_phi
def prime_factors(number):
primes = sieve.sieve(number)
factors = []
for prime in primes:
if number in primes and number not in factors:
factors.append(number)
break
if number % prime == 0:
factors.append(prime)
while number % prime == 0:
number = number / prime
return factors
def prime_factors(num):
primes = sieve(num)
index = 0
prime_factor = []
number_check = num
while index != len(primes) and number_check != 1:
prime_num = primes[index]
if number_check % prime_num == 0:
prime_factor.append(prime_num)
number_check /= prime_num
else:
index += 1
return prime_factor
def test_five():
assert sieve(1000) == [
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
]
def test_primes(self):
expected = [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]
self.assertEqual(expected, sieve(1000))
def pb35():
primes = set(sieve(10**6))
circular_count = 0
for p in primes:
circular = True
for r in number_rotations(p):
if r not in primes:
circular = False
break
if circular:
circular_count+=1
print circular_count
def prime_factors(num):
primes = sieve(num)
prime_factors_list = []
for prime in primes:
while num % prime == 0:
prime_factors_list.append(prime)
num = num // prime
if num in primes:
prime_factors_list.append(num)
break
else:
continue
prime_factors_list.sort()
return prime_factors_list
def primes3_2(pf, n):
if pf <= 0:
raise ValueError
elif n <= 2**pf:
return []
P = sieve(-(-n >> (pf-1))) # nを2**(pf-1)で割って小数点以下切り上げ
if pf == 1:
return P
PF = []
appendPF = PF.append
func1 = lambda x: (P[x]**2 < -(-n >> (pf-2)))
func2 = lambda x: (P[x]*p < -(-n >> (pf-2)))
high = len(P)
for j, p in enumerate(binary_takewhile(P, func1)):
high = upper_bound(j, high, func2)
appendPF(P[j:high] * p)
func1 = lambda x: (PF[x][0]*P[x] < sup)
func2 = lambda x: (PF[j][x]*p < sup)
func3 = lambda x: (PF[x][0]*p < sup)
func4 = lambda x: (S[x]*p < sup)
high1 = len(PF)
for i in range(3, pf+1):
sup = -(-n >> (pf-i))
high1 = upper_bound(0, high1, func1)
high3 = len(PF)
for j, p in enumerate(P[:high1]):
PF[j] = [binary_takewhile(PF[j], func2) * p]
high3 = upper_bound(j+1, high3, func3)
for S in PF[j+1:high3]:
PF[j].append(binary_takewhile(S, func4) * p)
PF[j] = np.concatenate(PF[j])
PF[j].sort()
PF = PF[:high1].copy()
del P
PF = np.concatenate(PF)
PF.sort()
return PF
def primes3(pf, n):
if pf <= 0:
raise ValueError
elif n <= 2**pf:
return []
P = sieve(-(-n >> (pf - 1))) # 小数点以下切り上げ
if pf == 1:
return P
PF = P.copy()
func = lambda x: (PF[x] * p < -(-n >> (pf - i)))
for i in range(2, pf + 1):
ip = 2**i
T = np.zeros(n - ip, dtype=np.bool)
for p in P:
T[(binary_takewhile(PF, func) * p) - ip] = True
PF = np.add(np.nonzero(T)[0].astype(np.uint64), np.uint64(ip))
del T
return PF
def primes4_2(pf, n):
if pf <= 0:
raise ValueError
elif n <= 2**pf:
return []
P = sieve(-(-n >> (pf-1)))
if pf == 1:
return P
PH = [0] * (pf-1)
PF = []
appendPF = PF.append
npprod = np.prod
high = len(P)
func = lambda x: (Pprod*P[x] < n)
while True:
index = PH[-1]
Pprod = npprod(P[PH])
if Pprod * P[index] < n:
high = upper_bound(index, high, func)
appendPF(P[index:high] * Pprod)
PH[-1] += 1
else:
for i in range(-2, -pf, -1):
if PH[i] != index:
PH[i:] = [PH[i] + 1] * (-i)
high = len(P)
break
else:
break
del P
PF = np.concatenate(PF)
PF.sort()
return PF
def quadSieve(n):
factor = 1
upperSieveBound = 50
while factor == 1 or factor == n:
upperSieveBound *= 2
sieveResult = sieve.sieve(n, upperSieveBound)
smoothNumbers = sieveResult[0]
primes = sieveResult[1]
matrix = []
for numTup in smoothNumbers:
expVec = [i % 2 for i in numTup[2]]
matrix.append(expVec)
if len(matrix) == 0:
continue
solutions = linearSolverMod2.solve(matrix)
for solution in solutions:
sieveExpVec = [0] * len(primes)
originalSquare = 1
for index in solution:
sieveExpVec = [
x + y for x, y in zip(sieveExpVec, smoothNumbers[index][2])
]
originalSquare *= smoothNumbers[index][0]
for i in range(len(sieveExpVec)):
sieveExpVec[i] //= 2
sieveVal = 1
for i in range(len(primes)):
sieveVal *= pow(primes[i], sieveExpVec[i])
factor = gcd(abs(originalSquare - sieveVal), n)
return {factor, n // factor}
def test_a_few_primes(self):
expected = [2, 3, 5, 7]
self.assertEqual(expected, sieve(10))
def main():
print(sieve(200000)[10001])
from sieve import sieve
from itertools import takewhile
# project Euler #10
primes = sieve(5000000)
if primes[-1] < 2000000:
raise Exception("too small: " + str(primes[-1]))
result = sum(takewhile(lambda x: x < 2000000, primes))
print(result)
# -*- coding: utf-8 -*-
"""
Created on Wed Jul 02 11:22:19 2014
@author: Milad
"""
import sieve
m = sieve.sieve(9999)
n = sieve.sieve(1000)
l = [x for x in m if x not in n]
d = {}
for i in l:
d[i] = "".join(sorted([x for x in str(i)]))
p = {}
for k, v in d.iteritems():
for l, w in d.iteritems():
if v == w and k != l:
if v in p.keys():
p[v].append(l)
p[v] = list(set(p[v]))
else:
p[v] = [k, l]
for k, v in p.iteritems():
v = sorted(v)
for i in range(len(v) - 2):
def test_pos(self):
"""Tests filtering of positive even numbers"""
self.assertEqual(sieve(POS, 2), [1, 3, 5, 7, 9])
def test_sieve(self):
self.assertEqual(sieve(10), [1,2,3,5,7,9])
def test_no_primes_under_two(self):
self.assertEqual(sieve(1), [])
def test_limit_is_prime(self):
self.assertEqual(sieve(13), [2, 3, 5, 7, 11, 13])
def test_find_primes_up_to_10(self):
self.assertEqual(sieve(10), [2, 3, 5, 7])
def test_find_first_prime(self):
self.assertEqual(sieve(2), [2])
def test_prime_limit(self):
expected = [2, 3, 5, 7]
self.assertEqual(expected, sieve(7))
import sieve
limit = 1000000
primes = sieve.sieve(limit)
# Create a list of consecutive sums of all primes up to one million
consec = [0]
for i in range(2, len(primes)):
if primes[i] == 1: # it is a prime number
temp = i + consec[-1]
if temp > limit * 10:
break
consec.append(temp)
# consec = consec[1:] # Remove zero from the start
print(consec)
# Index position in consec tells us how many primes are used to get that consec sum
# If want a consec sum from a starting position other than i = 0, then need to subtract the value
# preceeding the start position
# Adopt a method that works for lim = 100, must return 41 with 6 primes in the sum
answer = 0
max_consec = 0
for i in range(len(consec) - 1):
for j in range(i + 1, len(consec)):
consec_sum = consec[j] - consec[i]
if consec_sum >= limit:
break
if primes[consec_sum] == 1:
length = j - i
if length > max_consec:
max_consec = length
import sieve
# Composites in the list are marked as zero
primes = sieve.sieve(1000000)
# Main body of the problem, find how to break n into a prime and remainder
# The remainder must be even, otherwise cannot be 2*square
def check_conjecture(n):
i = 1
while 2 * i * i + 2 < n:
square = i * i
if primes[n - 2 * square] == 1: # is prime that conjecture true
return True
i += 1
return False
# Go over all odd composites
for i in range(9, len(primes), 2):
if primes[i] == 1: # not composite, skip
continue
if check_conjecture(i) == False:
print("\nAnswer: ", i)
break