# Sieve of Eratosthenes: making it quicker

I was thinking about doing problem 23 of Project Euler. It includes the difficulty where you have to factorize primes. I had already done that in problems I solved earlier, but it was only necessary to have a prime list until a relative small number.

So I had already implemented the algorithm when I took a look at this problem. I stored numbers in a list, flagged the ones that weren't prime numbers and removed them from the list, al of this in Python. It worked well for small lists, but when I went big, the memory usage was just too much.

I still wanted to use the algorithm of Eratosthenes, but I still needed a solution to my memory issue. So I decided, I don't know if it is the best solution, to write the list to a file and to constantly rewrite this file.

I succeeded in making this algorithm, but it's not really fast. To find primes up to the number 20 000 it took my program 405 seconds, or 6'45".

Is there any way I can improve this implementation? No file rewriting, cashing data in the RAM, etc.?

SieveCash.py (Python 3)

import tempfile
import os
import sys

def createSieve(outputFile, maxp, logging=0):
"""Creates a sieve and stores one prime per line in file outputFile."""
# Write a list of all numbers to check to a file
myNumFile = createBeginNumberList(maxp, outputFile)

# Do the looping
rfh = open(myNumFile, "r")
try:
for i in range(maxp+1):
# log if necessary
if logging > 0 and i % logging == 0:
print("Checking number", i, "(" + str(int(i/(maxp+1)*100)) + "%)")

# do thingy
break

if myNum != 0:
rfh.close()
rfh = None

rfh = open(myNumFile, "r")

finally:
if rfh != None:
rfh.close()

# Remove zeros
removeZerosInFile(outputFile)

return myNumFile

def createBeginNumberList(maxn, fileName):
"""@return value: the file path"""

tfh = open(fileName, "w")
try:
writeLineToFile(tfh, "0")
writeLineToFile(tfh, "0")
for i in range(2, maxn+1):
writeLineToFile(tfh, str(i))
finally:
tfh.close()

return fileName

def rewriteFileWithoutMultiples(inputFilename, multiple, startPos):
rfh = open(inputFilename, "r")
try:
writeFile = tempfile.mkstemp()
os.close(writeFile)
wfh = open(writeFile, "w")
except:
rfh.close()
raise

# Rewrite file
try:
writeLineToFile(wfh, str(aNum))

while True:
if aNum == None:
break

if aNum % multiple == 0:
writeLineToFile(wfh, "0")
else:
writeLineToFile(wfh, str(aNum))
finally:
rfh.close()
wfh.close()

# Copy
copyFile(writeFile, inputFilename)

def copyFile(source, dest):
rfh = open(source, "r")
try:
wfh = open(dest, "w")
except:
rfh.close()
raise

try:
while len(aLine) != 0:
wfh.write(aLine)
finally:
rfh.close()
wfh.close()

def removeZerosInFile(inputFile):
rfh = open(inputFile, "r")
try:
writeFile = tempfile.mkstemp()
os.close(writeFile)
wfh = open(writeFile, "w")
except:
rfh.close()
raise

# Rewrite file
try:
while True:
if aNum == None:
break

if aNum != 0:
writeLineToFile(wfh, str(aNum))

finally:
rfh.close()
wfh.close()

# Copy
copyFile(writeFile, inputFile)

def writeLineToFile(fh, l):
try:
fh.write(l)
fh.write('\n')
except:
fh.close()
raise

numb = len(mynum)
mynum = mynum.strip('\n')

if len(mynum) == 0:
return (0, None)

return (numb, int(mynum))

if __name__ == "__main__":
if len(sys.argv) < 2:
print("Usage:", sys.argv, "MAX_PRIME [LOG_INTERVAL]")
sys.exit(1)

maxp = int(sys.argv)
logging = int(maxp / 100)
try:
logging = int(sys.argv)
except:
pass

print("Creating seive until number", maxp, "with logging interval", logging)
createSieve(os.path.expanduser("~/Desktop/Sieve-" + str(maxp) + ".txt"),
maxp, logging)


Program Output (On an iMac 2.16 GHz Intel Core 2 Duo, 1GB RAM)

imac-van-ief2:23 ief2$SECONDS=0; python3 SieveCash.py 20000 2000; echo$SECONDS
Creating seive until number 20000 with logging interval 2000
Checking number 0 (0%)
Checking number 2000 (9%)
Checking number 4000 (19%)
Checking number 6000 (29%)
Checking number 8000 (39%)
Checking number 10000 (49%)
Checking number 12000 (59%)
Checking number 14000 (69%)
Checking number 16000 (79%)
Checking number 18000 (89%)
Checking number 20000 (99%)
405

Writing to disk is going to be really slow. Don't do that.

If you are running out of memory, try using the standard array module or the numpy library. Both of those provide efficient arrays for holding large numbers of values. Your only running out of memory because python's lists don't store their values in a memory efficient manner.

If you are running out of memory for some reason even taking that into account, post that code here and we'll be glad to point out how to improve the memory usage.

If space is the problem, a bit-array may help. Straightforward and unoptimised (I don't know Python so well),

import sys
import array

def sieve(n):
sieveBits = (n-1) // 2
sieveInts = (sieveBits+31) // 32
sieveBound = int(n**0.5) // 2
arr = array.array('I')
arr.extend((0,)*sieveInts)
for i in xrange(sieveBound):
if (arr[i >> 5] & (1 << (i&31))) == 0:
for j in xrange(2*i*(i+3)+3,sieveBits,2*i+3):
arr[j >> 5] |= 1 << (j&31)
return arr

def primes(n):
arr = sieve(n)
primes =  + [2*i+3 for i in xrange((n-1)//2) if arr[i >> 5] & (1 << (i & 31)) == 0]
return primes

if __name__ == "__main__":
if len(sys.argv) > 1:
up = int(sys.argv)
else:
up = 100000
print len(primes(up))


It's dog-slow (4.6 seconds to sieve up to 10,000,000), but the sieving doesn't take much memory.

• Yeah… my advice was basically going to be to use a typed array. :) That speeds up things quite a bit as it removed the equivalent of millions of “if” statements. Dec 4, 2011 at 12:08

Here is my version of the sieve. Ideone finds the 2262 primes less than 20000 in 0.02 seconds.

def sieve(n):
m = (n-1) // 2
b = [True]*m
i,p,ps = 0,3,
while p*p < n:
if b[i]:
ps.append(p)
j = 2*i*i + 6*i + 3
while j < m:
b[j] = False
j = j + 2*i + 3
i+=1; p+=2
while i < m:
if b[i]:
ps.append(p)
i+=1; p+=2
return ps

• way faster DUDE! please explain ur code so that many user will learn Jun 12, 2014 at 6:43

One minor improvement is to use xrange instead of range. In your case, you don't need range, and xrange has better memory performance, which might make a little difference.

• Sorry, I should've mentioned I use Python 3
– Ief2
Dec 2, 2011 at 19:51
• OK, then nevermind. Definitely removing all file IO would help
– TJD
Dec 2, 2011 at 19:52
• That's my initial problem. If I try to loop through 20 000 numbers, my computer can't handle anymore!
– Ief2
Dec 2, 2011 at 19:53