Many primes, very fast. Uses primesieve.

primesieve, one of the fastest (if not the fastest) prime sieve implementaions available, is actively maintained by Kim Walisch.

It uses a segmented sieve of Eratosthenes with wheel factorization for a complexity of O(nloglogn) operations.

Regarding primesieve for C++:

primesieve generates the first 50,847,534 primes up to 10^9 in just 0.4 seconds on a single core of an Intel Core i7-920 2.66GHz, this is about 50 times faster than an ordinary C/C++ sieve of Eratosthenes implementation and about 10,000 times faster than trial-division. primesieve outperforms [Kim's] older ecprime (fastest from 2002 to 2010) by about 30 percent and also substantially outperforms primegen the fastest sieve of Atkin implementation on the web.

For comparison, on an Intel Core i7 2GHz, pyprimesieve populates an entire Python list of the first 50,847,534 primes in 1.40 seconds. It's expected that a Python implementation would be slower than C++ but, surprisingly, by only one second.

pyprimesieve outperforms all of the fastest prime sieving implementations for Python.

Time (ms) to generate the all primes below one million and iterate over them in Python:

pyprimesieve         2.79903411865
primesfrom2to        13.1568908691
primesfrom3to        13.5800838470
ambi_sieve           16.1600112915
rwh_primes2          38.7749671936
rwh_primes1          48.5658645630
rwh_primes           52.0040988922
sieve_wheel_30       59.3869686127
sieveOfEratosthenes  59.4990253448
ambi_sieve_plain     161.740064621
sieveOfAtkin         232.724905014
sundaram3            251.194953918

It can be seen here that pyprimesieve is 4.7 times faster than the fastest Python alternative using Numpy and 13.85 times faster than the fastest pure Python sieve.

All benchmark scripts and algorithms are available for reproduction. Prime sieve algorithm implementations were taken from this discussion on SO.

primes(n): List of prime numbers up to n.

primes(start, n): List of prime numbers from start up to n.

primes_sum(n): The summation of prime numbers up to n. The optimal number of threads will be determined for the given number and system.

primes_sum(start, n): The summation of prime numbers from start up to n. The optimal number of threads will be determined for the given numbers and system.

primes_nth(n): The nth prime number.

factorize(n): List of tuples in the form of (prime, power) for the prime factorization of n.

python setup.py install

NOTE: Because of the need to use OpenMP to compile the parallelized version of summation, g++ is specified in environment variables of setup to avoid distutils choosing a compiler that does not have support for OpenMP. If you don't have g++, you will need to change that in setup.py.

Like any C/C++ extension, you need to have the development package of Python (able to include Python.h) in order to compile. On Ubuntu you can simply run,

sudo apt-get install python-dev

After installation, you can make sure everything is working by running the following inside the project root folder,

python tests

"Modified BSD License". See LICENSE for details. Copyright Jared Suttles, 2013.