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1) for n in range(limit + 1): if is_prime[n]: yield n for i in range(n*n, limit+1, n): # start at ``n`` squared is_prime[i] = False @tag('pure') def iprimes_upto_enumerate(limit): is_prime = [False] * 2 + [True] * (limit - 1) for n, p in enumerate(is_prime): if p: yield n for i in range(n*n, limit+1, n): # start at ``n`` squared is_prime[i] = False try: import math import numpy @tag('sequence', 'numpy') def nolfonzo_prime6(upto): # http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3638617#3638617 primes=numpy.arange(3,upto+1,2) isprime=numpy.ones((upto-1)/2,dtype=bool) for factor in primes[:int(math.sqrt(upto))]: if isprime[(factor-2)/2]: isprime[(factor*3-2)/2:(upto-1)/2:factor]=0 return numpy.insert(primes[isprime],0,2) # http://rosettacode.org/wiki/Sieve_of_Eratosthenes#Using_numpy import numpy @tag('sequence', 'numpy') def primes_upto2(limit): is_prime = numpy.ones(limit + 1, dtype=numpy.bool) for n in xrange(2, int(limit**0.5 + 1.5)): if is_prime[n]: is_prime[n*n::n] = 0 return numpy.nonzero(is_prime)[0][2:] @tag('sequence', 'numpy') def primes_upto4(limit): is_prime = numpy.ones(limit + 1, dtype=numpy.bool) for n in xrange(3, int(limit**0.5 + 1.5), 2): if is_prime[n]: is_prime[n*n::2*n] = 0 is_prime[:2] = is_prime[4::2] = 0 return is_prime.nonzero()[0] @tag('numpy') def primes_upto4_gen(limit): is_prime = numpy.ones(limit + 1, dtype=numpy.bool) for n in xrange(3, int(limit**0.5 + 1.5), 2): if is_prime[n]: is_prime[n*n::2*n] = 0 yield 2 for i, p in enumerate(is_prime[3::2]): if p: yield 3 + 2*i @tag('numpy') def primes_upto2_gen(limit): is_prime = numpy.ones(limit + 1, dtype=numpy.bool) for n in xrange(2, int(limit**0.5 + 1.5)): if is_prime[n]: is_prime[n*n::n] = 0 return (i for i in xrange(2, limit+1) if is_prime[i]) @tag('numpy') def primes_upto2_gen2(limit): """Generate prime numbers less than limit. Assume limit > 2 """ yield 2 is_prime = numpy.ones(limit, dtype=numpy.bool) for n in xrange(3, int(limit**.5)+1, 2): if is_prime[n]: yield n is_prime[n*n::n] = False for n in xrange(n+2, limit, 2): if is_prime[n]: yield n @tag('numpy') def primes_upto2_gen3(limit): """Generate prime numbers less than limit. Assume limit > 2 """ yield 2 is_prime = numpy.ones(limit, dtype=numpy.bool) for n in xrange(3, int(limit**.5)+1, 2): if is_prime[n]: yield n is_prime[n*n::2*n] = False for n in xrange(n+2, limit, 2): if is_prime[n]: yield n from numpy import array, bool_, multiply, nonzero, ones, put, resize # def makepattern(smallprimes): pattern = ones(multiply.reduce(smallprimes), dtype=bool_) pattern[0] = 0 for p in smallprimes: pattern[p::p] = 0 return pattern # @tag('sequence', 'numpy') def primes_upto3(limit, smallprimes=(2,3,5,7,11)): sp = array(smallprimes) if limit <= sp.max(): return sp[sp <= limit] # isprime = resize(makepattern(sp), limit + 1) isprime[:2] = 0; put(isprime, sp, 1) # for n in range(sp.max() + 2, int(limit**0.5 + 1.5), 2): if isprime[n]: isprime[n*n::n] = 0 return nonzero(isprime)[0] @tag('sequence', 'numpy') def primesfrom3to(n): """ Returns a array of primes, 3 <= p < n """ # http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188 sieve = numpy.ones(n/2, dtype=numpy.bool) for i in xrange(3,int(n**0.5)+1,2): if sieve[i/2]: sieve[i*i/2::i] = False return numpy.r_[2,2*numpy.nonzero(sieve)[0][1::]+1] @tag('sequence', 'numpy') def primesfrom2to(n): """ Input n>=6, Returns a array of primes, 2 <= p < n """ sieve = numpy.ones(n/3+((n+1)&1)*(n%3)/2, dtype=numpy.bool) sieve[0] = False for i in xrange(int(n**0.5)/3+1): if sieve[i]: k=3*i+1+(i&1) sieve[ ((k*k)/3) ::2*k] = False sieve[(k*k+4*k-2*k*(i&1))/3::2*k] = False primes = 3*numpy.nonzero(sieve)[0]+1 primes += (primes+1)%2 return numpy.r_[2,3,primes] @tag('sequence', 'numpy') def primesfrom2to2(n): """ Input n>=6, Returns a array of primes, 2 <= p < n """ sieve = numpy.ones(n/3+((n+1)&1)*(n%3)/2, dtype=numpy.bool) sieve[0] = False for i in xrange(int(n**0.5)/3+1): if sieve[i]: k=3*i+1+(i&1) sieve[ ((k*k)/3) ::2*k] = False sieve[(k*k+4*k-2*k*(i&1))/3::2*k] = False return numpy.r_[2,3,((3*numpy.nonzero(sieve)[0]+1)|1)] @tag('sequence', 'numpy') def primesfrom2to3(n): """ Input n>=6, Returns a array of primes, 2 <= p < n """ sieve = numpy.ones(n/3 + (n%6==2), dtype=numpy.bool) sieve[0] = False for i in xrange(int(n**0.5)/3+1): if sieve[i]: k=3*i+1|1 sieve[ ((k*k)/3) ::2*k] = False sieve[(k*k+4*k-2*k*(i&1))/3::2*k] = False return numpy.r_[2,3,((3*numpy.nonzero(sieve)[0]+1)|1)] @tag('numpy') def primesgen1(n): """ Generates all primes < n """ # http://stackoverflow.com/questions/2897297/speed-up-bitstring-bit-operations-in-python/2908512#2908512 sieve1 = numpy.ones(n/6+1, dtype=numpy.bool) sieve5 = numpy.ones(n/6 , dtype=numpy.bool) sieve1[0] = False yield 2; yield 3; for i in xrange(int(n**0.5)/6+1): if sieve1[i]: k=6*i+1; yield k; sieve1[ ((k*k)/6) ::k] = False sieve5[(k*k+4*k)/6::k] = False if sieve5[i]: k=6*i+5; yield k; sieve1[ ((k*k)/6) ::k] = False sieve5[(k*k+2*k)/6::k] = False for i in xrange(i+1,n/6): if sieve1[i]: yield 6*i+1 if sieve5[i]: yield 6*i+5 if 1 < n%6 <= 5: if sieve1[i+1]: yield 6*i+1 @tag('numpy') def primesgen2(n): # http://stackoverflow.com/questions/2897297/speed-up-bitstring-bit-operations-in-python/2908512#2908512 """ Input n>=30, Generates all primes < n """ sieve01 = numpy.ones(n/30+1, dtype=numpy.bool) sieve07 = numpy.ones(n/30+1, dtype=numpy.bool) sieve11 = numpy.ones(n/30+1, dtype=numpy.bool) sieve13 = numpy.ones(n/30+1, dtype=numpy.bool) sieve17 = numpy.ones(n/30+1, dtype=numpy.bool) sieve19 = numpy.ones(n/30+1, dtype=numpy.bool) sieve23 = numpy.ones(n/30+1, dtype=numpy.bool) sieve29 = numpy.ones(n/30 , dtype=numpy.bool) sieve01[0] = False yield 2; yield 3; yield 5; for i in xrange(int(n**0.5)/30+1): if sieve01[i]: k=30*i+1; yield k; sieve01[ (k*k)/30::k] = False sieve07[(k*k+ 6*k)/30::k] = False sieve11[(k*k+10*k)/30::k] = False sieve13[(k*k+12*k)/30::k] = False sieve17[(k*k+16*k)/30::k] = False sieve19[(k*k+18*k)/30::k] = False sieve23[(k*k+22*k)/30::k] = False sieve29[(k*k+28*k)/30::k] = False if sieve07[i]: k=30*i+7; yield k; sieve01[(k*k+ 6*k)/30::k] = False sieve07[(k*k+24*k)/30::k] = False sieve11[(k*k+16*k)/30::k] = False sieve13[(k*k+12*k)/30::k] = False sieve17[(k*k+ 4*k)/30::k] = False sieve19[ (k*k)/30::k] = False sieve23[(k*k+22*k)/30::k] = False sieve29[(k*k+10*k)/30::k] = False if sieve11[i]: k=30*i+11; yield k; sieve01[ (k*k)/30::k] = False sieve07[(k*k+ 6*k)/30::k] = False sieve11[(k*k+20*k)/30::k] = False sieve13[(k*k+12*k)/30::k] = False sieve17[(k*k+26*k)/30::k] = False sieve19[(k*k+18*k)/30::k] = False sieve23[(k*k+ 2*k)/30::k] = False sieve29[(k*k+ 8*k)/30::k] = False if sieve13[i]: k=30*i+13; yield k; sieve01[(k*k+24*k)/30::k] = False sieve07[(k*k+ 6*k)/30::k] = False sieve11[(k*k+ 4*k)/30::k] = False sieve13[(k*k+18*k)/30::k] = False sieve17[(k*k+16*k)/30::k] = False sieve19[ (k*k)/30::k] = False sieve23[(k*k+28*k)/30::k] = False sieve29[(k*k+10*k)/30::k] = False if sieve17[i]: k=30*i+17; yield k; sieve01[(k*k+ 6*k)/30::k] = False sieve07[(k*k+24*k)/30::k] = False sieve11[(k*k+26*k)/30::k] = False sieve13[(k*k+12*k)/30::k] = False sieve17[(k*k+14*k)/30::k] = False sieve19[ (k*k)/30::k] = False sieve23[(k*k+ 2*k)/30::k] = False sieve29[(k*k+20*k)/30::k] = False if sieve19[i]: k=30*i+19; yield k; sieve01[ (k*k)/30::k] = False sieve07[(k*k+24*k)/30::k] = False sieve11[(k*k+10*k)/30::k] = False sieve13[(k*k+18*k)/30::k] = False sieve17[(k*k+ 4*k)/30::k] = False sieve19[(k*k+12*k)/30::k] = False sieve23[(k*k+28*k)/30::k] = False sieve29[(k*k+22*k)/30::k] = False if sieve23[i]: k=30*i+23; yield k; sieve01[(k*k+24*k)/30::k] = False sieve07[(k*k+ 6*k)/30::k] = False sieve11[(k*k+14*k)/30::k] = False sieve13[(k*k+18*k)/30::k] = False sieve17[(k*k+26*k)/30::k] = False sieve19[ (k*k)/30::k] = False sieve23[(k*k+ 8*k)/30::k] = False sieve29[(k*k+20*k)/30::k] = False if sieve29[i]: k=30*i+29; yield k; sieve01[ (k*k)/30::k] = False sieve07[(k*k+24*k)/30::k] = False sieve11[(k*k+20*k)/30::k] = False sieve13[(k*k+18*k)/30::k] = False sieve17[(k*k+14*k)/30::k] = False sieve19[(k*k+12*k)/30::k] = False sieve23[(k*k+ 8*k)/30::k] = False sieve29[(k*k+ 2*k)/30::k] = False for i in xrange(i+1,n/30): if sieve01[i]: yield 30*i+1 if sieve07[i]: yield 30*i+7 if sieve11[i]: yield 30*i+11 if sieve13[i]: yield 30*i+13 if sieve17[i]: yield 30*i+17 if sieve19[i]: yield 30*i+19 if sieve23[i]: yield 30*i+23 if sieve29[i]: yield 30*i+29 if n%30 > 1: if sieve01[i+1]: yield 30*i+1 if n%30 > 7: if sieve07[i+1]: yield 30*i+7 if n%30 > 11: if sieve11[i+1]: yield 30*i+11 if n%30 > 13: if sieve13[i+1]: yield 30*i+13 if n%30 > 17: if sieve17[i+1]: yield 30*i+17 if n%30 > 19: if sieve19[i+1]: yield 30*i+19 if n%30 > 23: if sieve23[i+1]: yield 30*i+23 except ImportError: pass #http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188 @tag('pure', 'sequence') def primes1(n): """ Returns a list of primes < n """ sieve = [True] * (n//2) for i in range(3,int(n**0.5)+1,2): if sieve[i//2]: sieve[i*i//2::i] = [False] * ((n-i*i-1)//(2*i)+1) return [2] + [2*i+1 for i in range(1,n//2) if sieve[i]] @tag('pure', 'sequence') def primes(n): """ Returns a list of primes < n """ sieve = [True] * n for i in range(3,int(n**0.5)+1,2): if sieve[i]: sieve[i*i::2*i]=[False]*((n-i*i-1)//(2*i)+1) return [2] + [i for i in range(3,n,2) if sieve[i]] try: #http://stackoverflow.com/questions/2897297/speed-up-bitstring-bit-operations-in-python/2901856#2901856 import gmpy2 @tag('gmpy2') def prime_numbers(limit=1000000): '''Prime number generator. Yields the series 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ... using Sieve of Eratosthenes. ''' yield 2 sub_limit = int(limit**0.5) # Actual number is 2*bit_position + 1. oddnums = gmpy2.xmpz(1) current = 0 while True: current += 1 current = oddnums.bit_scan0(current) prime = 2 * current + 1 if prime > limit: break yield prime # Exclude further multiples of the current prime number if prime <= sub_limit: for j in range(2*current*(current+1), limit>>1, prime): oddnums.bit_set(j) @tag('gmpy2') def prime_numbers2(limit=1000000): '''Prime number generator. Yields the series 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ... using Sieve of Eratosthenes. ''' yield 2 sub_limit = int(limit**0.5) # Actual number is 2*bit_position + 1. oddnums = gmpy2.xmpz(1) f_set = oddnums.bit_set f_scan0 = oddnums.bit_scan0 current = 0 while True: current += 1 current = f_scan0(current) prime = 2 * current + 1 if prime > limit: break yield prime # Exclude further multiples of the current prime number if prime <= sub_limit: consume(map(f_set,range(2*current*(current+1), limit>>1, prime))) @tag('gmpy2') def prime_numbers4(limit=1000000): '''Prime number generator. Yields the series 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ... using Sieve of Eratosthenes. ''' sub_limit = int(limit**0.5) flags = gmpy2.xmpz(1) flags[(limit>>1)+1] = True f_scan0 = flags.bit_scan0 current = 0 prime = 2 while prime <= sub_limit: yield prime current += 1 current = f_scan0(current) prime = 2 * current + 1 flags[2*current*(current+1):limit>>1:prime] = True while prime <= limit: yield prime current += 1 current = f_scan0(current) prime = 2 * current + 1 # @tag('gmpy2') def gmpy_next_prime_primes(limit): next_ = gmpy2.next_prime p = next_(1) while p < limit: yield p p = next_(p) except ImportError: pass ############################################################################### #http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/2068548#2068548 from math import sqrt, ceil try: import builtins except ImportError: import __builtin__ as builtins # small primes array __smallp = ( 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) @tag('pure', 'sequence') def sieve_wheel_30(N): # http://zerovolt.com/?p=88 ''' Returns a list of primes <= N using wheel criterion 2*3*5 = 30 Copyright 2009 by zerovolt.com This code is free for non-commercial purposes, in which case you can just leave this comment as a credit for my work. If you need this code for commercial purposes, please contact me by sending an email to: info [at] zerovolt [dot] com.''' wheel = (2, 3, 5) const = 30 if N < 2: return [] if N <= const: pos = 0 while __smallp[pos] <= N: pos += 1 return list(__smallp[:pos]) # make the offsets list offsets = (7, 11, 13, 17, 19, 23, 29, 1) # prepare the list p = [2, 3, 5] dim = 2 + N // const tk1 = [True] * dim tk7 = [True] * dim tk11 = [True] * dim tk13 = [True] * dim tk17 = [True] * dim tk19 = [True] * dim tk23 = [True] * dim tk29 = [True] * dim tk1[0] = False # help dictionary d # d[a , b] = c ==> if I want to find the smallest useful multiple of (30*pos)+a # on tkc, then I need the index given by the product of [(30*pos)+a][(30*pos)+b] # in general. If b < a, I need [(30*pos)+a][(30*(pos+1))+b] d = {} for x in offsets: for y in offsets: res = (x*y) % const if res in offsets: d[(x, res)] = y # another help dictionary: gives tkx calling tmptk[x] tmptk = {1:tk1, 7:tk7, 11:tk11, 13:tk13, 17:tk17, 19:tk19, 23:tk23, 29:tk29} pos, prime, lastadded, stop = 0, 0, 0, int(ceil(sqrt(N))) # inner functions definition def del_mult(tk, start, step): for k in range(start, len(tk), step): tk[k] = False # end of inner functions definition cpos = const * pos while prime < stop: # 30k + 7 if tk7[pos]: prime = cpos + 7 p.append(prime) lastadded = 7 for off in offsets: tmp = d[(7, off)] start = (pos + prime) if off == 7 else (prime * (const * (pos + 1 if tmp < 7 else 0) + tmp) )//const del_mult(tmptk[off], start, prime) # 30k + 11 if tk11[pos]: prime = cpos + 11 p.append(prime) lastadded = 11 for off in offsets: tmp = d[(11, off)] start = (pos + prime) if off == 11 else (prime * (const * (pos + 1 if tmp < 11 else 0) + tmp) )//const del_mult(tmptk[off], start, prime) # 30k + 13 if tk13[pos]: prime = cpos + 13 p.append(prime) lastadded = 13 for off in offsets: tmp = d[(13, off)] start = (pos + prime) if off == 13 else (prime * (const * (pos + 1 if tmp < 13 else 0) + tmp) )//const del_mult(tmptk[off], start, prime) # 30k + 17 if tk17[pos]: prime = cpos + 17 p.append(prime) lastadded = 17 for off in offsets: tmp = d[(17, off)] start = (pos + prime) if off == 17 else (prime * (const * (pos + 1 if tmp < 17 else 0) + tmp) )//const del_mult(tmptk[off], start, prime) # 30k + 19 if tk19[pos]: prime = cpos + 19 p.append(prime) lastadded = 19 for off in offsets: tmp = d[(19, off)] start = (pos + prime) if off == 19 else (prime * (const * (pos + 1 if tmp < 19 else 0) + tmp) )//const del_mult(tmptk[off], start, prime) # 30k + 23 if tk23[pos]: prime = cpos + 23 p.append(prime) lastadded = 23 for off in offsets: tmp = d[(23, off)] start = (pos + prime) if off == 23 else (prime * (const * (pos + 1 if tmp < 23 else 0) + tmp) )//const del_mult(tmptk[off], start, prime) # 30k + 29 if tk29[pos]: prime = cpos + 29 p.append(prime) lastadded = 29 for off in offsets: tmp = d[(29, off)] start = (pos + prime) if off == 29 else (prime * (const * (pos + 1 if tmp < 29 else 0) + tmp) )//const del_mult(tmptk[off], start, prime) # now we go back to top tk1, so we need to increase pos by 1 pos += 1 cpos = const * pos # 30k + 1 if tk1[pos]: prime = cpos + 1 p.append(prime) lastadded = 1 for off in offsets: tmp = d[(1, off)] start = (pos + prime) if off == 1 else (prime * (const * pos + tmp) )//const del_mult(tmptk[off], start, prime) # time to add remaining primes # if lastadded == 1, remove last element and start adding them from tk1 # this way we don't need an "if" within the last while if lastadded == 1: p.pop() # now complete for every other possible prime while pos < len(tk1): cpos = const * pos if tk1[pos]: p.append(cpos + 1) if tk7[pos]: p.append(cpos + 7) if tk11[pos]: p.append(cpos + 11) if tk13[pos]: p.append(cpos + 13) if tk17[pos]: p.append(cpos + 17) if tk19[pos]: p.append(cpos + 19) if tk23[pos]: p.append(cpos + 23) if tk29[pos]: p.append(cpos + 29) pos += 1 # remove exceeding if present pos = len(p) - 1 while p[pos] > N: pos -= 1 if pos < len(p) - 1: del p[pos+1:] # return p list return p @tag('pure', 'sequence') def sieveOfEratosthenes(n): """sieveOfEratosthenes(n): return the list of the primes < n.""" # Code from: <dickinsm@gmail.com>, Nov 30 2006 # http://groups.google.com/group/comp.lang.python/msg/f1f10ced88c68c2d if n <= 2: return [] sieve = range(3, n, 2) if type(sieve) != list: sieve = list(sieve) top = len(sieve) for si in sieve: if si: bottom = (si*si - 3) // 2 if bottom >= top: break sieve[bottom::si] = [0] * -((bottom - top) // si) return [2] + [el for el in sieve if el] @tag('pure', 'sequence') def sieveOfAtkin(end): """sieveOfAtkin(end): return a list of all the prime numbers <end using the Sieve of Atkin.""" # Code by Steve Krenzel, <Sgk284@gmail.com>, improved # Code: http://krenzel.info/?p=83 # Info: http://en.wikipedia.org/wiki/Sieve_of_Atkin assert end > 0, "end must be >0" lng = ((end-1) // 2) sieve = [False] * (lng + 1) x_max, x2, xd = int(sqrt((end-1)/4.0)), 0, 4 for xd in range(4, 8*x_max + 2, 8): x2 += xd y_max = int(sqrt(end-x2)) n, n_diff = x2 + y_max*y_max, (y_max << 1) - 1 if not (n & 1): n -= n_diff n_diff -= 2 for d in range((n_diff - 1) << 1, -1, -8): m = n % 12 if m == 1 or m == 5: m = n >> 1 sieve[m] = not sieve[m] n -= d x_max, x2, xd = int(sqrt((end-1) / 3.0)), 0, 3 for xd in range(3, 6 * x_max + 2, 6): x2 += xd y_max = int(sqrt(end-x2)) n, n_diff = x2 + y_max*y_max, (y_max << 1) - 1 if not(n & 1): n -= n_diff n_diff -= 2 for d in range((n_diff - 1) << 1, -1, -8): if n % 12 == 7: m = n >> 1 sieve[m] = not sieve[m] n -= d x_max, y_min, x2, xd = int((2 + sqrt(4-8*(1-end)))/4), -1, 0, 3 for x in range(1, x_max + 1): x2 += xd xd += 6 if x2 >= end: y_min = (((int(ceil(sqrt(x2 - end))) - 1) << 1) - 2) << 1 n, n_diff = ((x*x + x) << 1) - 1, (((x-1) << 1) - 2) << 1 for d in range(n_diff, y_min, -8): if n % 12 == 11: m = n >> 1 sieve[m] = not sieve[m] n += d primes = [2, 3] if end <= 3: return primes[:max(0,end-2)] for n in range(5 >> 1, (int(sqrt(end))+1) >> 1): if sieve[n]: primes.append((n << 1) + 1) aux = (n << 1) + 1 aux *= aux for k in range(aux, end, 2 * aux): sieve[k >> 1] = False s = int(sqrt(end)) + 1 if s % 2 == 0: s += 1 primes.extend([i for i in range(s, end, 2) if sieve[i >> 1]]) return primes try: import numpy as np @tag('sequence', 'numpy') def ambi_sieve(n): # http://tommih.blogspot.com/2009/04/fast-prime-number-generator.html s = np.arange(3, n, 2) for m in builtins.xrange(3, int(n ** 0.5)+1, 2): if s[(m-3)/2]: s[(m*m-3)/2::m]=0 return np.r_[2, s[s>0]] except ImportError: pass @tag('pure', 'sequence') def ambi_sieve_plain(n): s = list(range(3, n, 2)) for m in range(3, int(n**0.5)+1, 2): if s[(m-3)//2]: for t in range((m*m-3)//2,(n>>1)-1,m): s[t]=0 return [2]+[t for t in s if t>0] @tag('pure', 'sequence') def sundaram3(max_n): numbers = list(range(3, max_n+1, 2)) half = (max_n)//2 initial = 4 for step in range(3, max_n+1, 2): for i in range(initial, half, step): numbers[i-1] = 0 initial += 2*(step+1) if initial > half: return [2] + [n for n in numbers if n] ############################################################################### @tag('pure', 'sequence') def rwh_primes2(n): # http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188 """ Input n>=6, Returns a list of primes, 2 <= p < n """ n, correction = n-n%6+6, 2-(n%6>1) sieve = [True] * (n//3) for i in range(1,int(n**0.5)//3+1): if sieve[i]: k=3*i+1|1 sieve[ k*k//3 ::2*k] = [False] * ((n//6-k*k//6-1)//k+1) sieve[k*(k-2*(i&1)+4)//3::2*k] = [False] * ((n//6-k*(k-2*(i&1)+4)//6-1)//k+1) return [2,3] + [3*i+1|1 for i in range(1,n//3-correction) if sieve[i]] @tag('pure', 'sequence') def sieveOfErat_Noldorin(end): # http://stackoverflow.com/questions/1023768/sieve-of-atkin-explanation/1023777#1023777 if end < 2: return [] #The array doesn't need to include even numbers lng = ((end//2)-1+end%2) # Create array and assume all numbers in array are prime sieve = [True]*(lng+1) # In the following code, you're going to see some funky # bit shifting and stuff, this is just transforming i and j # so that they represent the proper elements in the array. # The transforming is not optimal, and the number of # operations involved can be reduced. # Only go up to square root of the end for i in range(int(sqrt(end)) >> 1): # Skip numbers that aren't marked as prime if not sieve[i]: continue # Unmark all multiples of i, starting at i**2 for j in range( (i*(i + 3) << 1) + 3, lng, (i << 1) + 3): sieve[j] = False # Don't forget 2! primes = [2] # Gather all the primes into a list, leaving out the composite numbers primes.extend([(i << 1) + 3 for i in range(lng) if sieve[i]]) return primes try: import primegen # http://cr.yp.to/primegen.html @tag('primegen') def primegen_upto(n): pg = primegen.primegen() primegen.primegen_init(pg) next_ = primegen.primegen_next p = next_(pg) while p < n: yield p p = next_(pg) except ImportError: pass @tag('pure') def eratosthenes_dugres(n): # http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3328618#3328618 N=list(range(n+1)) z=[0]*(n//2) for i in range(2, int(n**.5)+1): if N[i]: N[i*i::i] = z[:(n//i)-i+1] return filter(None, N[2:]) try: bytearray # `bytearray` is not present in pypy @tag('pure') def prime_pure(limit=1000000): # http://stackoverflow.com/questions/2897297/speed-up-bitstring-bit-operations-in-python/2897459#2897459 yield 2 flags = bytearray((limit + 7) // 8) sub_limit = int(limit**0.5) for i in range(3, limit, 2): byte, bit = divmod(i, 8) if not flags[byte] & (128 >> bit): yield i if i <= sub_limit: for j in range(i*3, limit, i*2): byte, bit = divmod(j, 8) flags[byte] |= (128 >> bit) except NameError: pass # ################################################################################ def consume(iterator): collections.deque(iterator, maxlen=0) @profile def test(n=1000000): nsmall = 10000 result = list(register.functions[0](nsmall)) for f in register.functions[1:]: for expected, got in zip(result, f(nsmall)): assert expected == got, (expected, got, f.__name__) results = [test_func(f, n, quiet=True) for f in register.functions] results.sort(key=lambda x:x[1]) for r in results: print25('%s:%s %s' % ( str(r[0]).ljust(25), ('%d ms' % (int(r[1]*1e3+0.5),)).rjust(10), r[2])) def test_func(f, n=1000000, quiet=False, repeat=3, ntimes=10): stmt = '%s(%d)' % (f.__name__, n) setup = 'from %s import %s' % (__name__, f.__name__) if not 'sequence' in f.tags: stmt = 'consume(%s)' % (stmt,) setup += '; from %s import consume' % (__name__,) t = timeit.Timer(stmt, setup) try: results = [r/ntimes for r in t.repeat(repeat=repeat, number=ntimes)] except Exception: time = -1 else: time = min(results) result = (f.__name__, time, ':'.join(f.tags)) if not quiet: print25(*result) return result if __name__=="__main__": test() try: for f in [primesgen2, primesfrom2to3]: test_func(f, n=10**9, repeat=1, ntimes=1) except NameError: pass # Python2.6 n=1e6 # primesfrom2to3 : 7 ms numpy:sequence # primesfrom2to2 : 7 ms numpy:sequence # ambi_sieve : 9 ms numpy:sequence # primesfrom2to : 9 ms numpy:sequence # primesfrom3to : 9 ms numpy:sequence # nolfonzo_prime6 : 12 ms numpy:sequence # primes_upto4 : 15 ms numpy:sequence # primes_upto3 : 15 ms numpy:sequence # primes_upto2 : 17 ms numpy:sequence # primegen_upto : 31 ms primegen # prime_numbers4 : 32 ms gmpy2 # primesgen2 : 35 ms numpy # primesgen1 : 38 ms numpy # rwh_primes2 : 44 ms pure:sequence # primes_upto2_gen3 : 47 ms numpy # primes_upto2_gen2 : 48 ms numpy # primes1 : 49 ms pure:sequence # primes : 50 ms pure:sequence # primes_upto4_gen : 53 ms numpy # sieve_wheel_30 : 56 ms pure:sequence # sieveOfEratosthenes : 79 ms pure:sequence # prime_numbers2 : 83 ms gmpy2 # primes_upto2_gen : 91 ms numpy # sieveOfErat_Noldorin : 112 ms pure:sequence # ambi_sieve_plain : 145 ms pure:sequence # prime_numbers : 146 ms gmpy2 # eratosthenes_dugres : 161 ms pure # sieveOfAtkin : 219 ms pure:sequence # sundaram3 : 235 ms pure:sequence # iprimes_upto : 281 ms pure # iprimes_upto_enumerate : 283 ms pure # prime_pure : 605 ms pure # Python2.6 n=1e9 # primesgen1 35.7735440731 numpy # primesgen2 27.4978778362 numpy # primesfrom2to3 9.91236019135 numpy:sequence # Python3 n=1e6 # prime_numbers4 : 32 ms gmpy2 # rwh_primes2 : 38 ms pure:sequence # sieve_wheel_30 : 44 ms pure:sequence # primes : 48 ms pure:sequence # primes1 : 49 ms pure:sequence # sieveOfEratosthenes : 49 ms pure:sequence # prime_numbers2 : 92 ms gmpy2 # sieveOfErat_Noldorin : 93 ms pure:sequence # ambi_sieve_plain : 109 ms pure:sequence # prime_numbers : 112 ms gmpy2 # eratosthenes_dugres : 112 ms pure # sundaram3 : 187 ms pure:sequence # sieveOfAtkin : 199 ms pure:sequence # iprimes_upto_enumerate : 225 ms pure # iprimes_upto : 227 ms pure # prime_pure : 500 ms pure # pypy n=1e6 # rwh_primes2 : 84 ms pure:sequence # sieve_wheel_30 : 113 ms pure:sequence # primes : 115 ms pure:sequence # primes1 : 116 ms pure:sequence # sieveOfEratosthenes : 135 ms pure:sequence # sieveOfErat_Noldorin : 251 ms pure:sequence # ambi_sieve_plain : 252 ms pure:sequence # sieveOfAtkin : 347 ms pure:sequence # sundaram3 : 436 ms pure:sequence |
"""Compare performance of differenct Sieve of Eratosthenes implementations."""
#NOTE: maintain single source 2.5-x and 3.x compatibility
# if `numpy`/`gmpy2` is installed test it
import collections, itertools, sys

import timeit

try:
    range = xrange
    map = itertools.imap
    filter = itertools.ifilter
except NameError:
    pass

def print25(*args, **kwargs):
    # `from __future__ import print_function` doesn't work on Python
    # <= 2.5
    sep  = kwargs.get('sep', ' ')
    end  = kwargs.get('end', '\n')
    file = kwargs.get('file', sys.stdout)
    file.write(sep.join(map(str, args)))
    file.write(end)

def profile(func):
    return func

if '--profile' in sys.argv:
    try:
        from profilestats import profile # pip install profilestats
    except ImportError:
        pass

def register(func):
    register.functions.append(func)
    return func
register.functions = []

def tag(*tags):
    def decorator(func):
        func.tags = getattr(func, 'tags', set())
        func.tags.update(tags)
        return register(func)
    return decorator

# http://rosettacode.org/wiki/Sieve_of_Eratosthenes#Using_generator
@tag('pure')
def iprimes_upto(limit):
    is_prime = [False] * 2 + [True] * (limit - 1)
    for n in range(limit + 1):
        if is_prime[n]:
            yield n
            for i in range(n*n, limit+1, n): # start at ``n`` squared
                is_prime[i] = False
@tag('pure')
def iprimes_upto_enumerate(limit):
    is_prime = [False] * 2 + [True] * (limit - 1)
    for n, p in enumerate(is_prime):
        if p:
            yield n
            for i in range(n*n, limit+1, n): # start at ``n`` squared
                is_prime[i] = False

try:
    import math
    import numpy
    @tag('sequence', 'numpy')
    def nolfonzo_prime6(upto):
        # http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3638617#3638617
        primes=numpy.arange(3,upto+1,2)
        isprime=numpy.ones((upto-1)/2,dtype=bool)
        for factor in primes[:int(math.sqrt(upto))]:
            if isprime[(factor-2)/2]: isprime[(factor*3-2)/2:(upto-1)/2:factor]=0
        return numpy.insert(primes[isprime],0,2)

    # http://rosettacode.org/wiki/Sieve_of_Eratosthenes#Using_numpy
    import numpy
    @tag('sequence', 'numpy')
    def primes_upto2(limit):
        is_prime = numpy.ones(limit + 1, dtype=numpy.bool)
        for n in xrange(2, int(limit**0.5 + 1.5)): 
            if is_prime[n]:
                is_prime[n*n::n] = 0
        return numpy.nonzero(is_prime)[0][2:]

    @tag('sequence', 'numpy')
    def primes_upto4(limit):
        is_prime = numpy.ones(limit + 1, dtype=numpy.bool)
        for n in xrange(3, int(limit**0.5 + 1.5), 2): 
            if is_prime[n]:
                is_prime[n*n::2*n] = 0        
        is_prime[:2] = is_prime[4::2] = 0
        return is_prime.nonzero()[0]

    @tag('numpy')
    def primes_upto4_gen(limit):
        is_prime = numpy.ones(limit + 1, dtype=numpy.bool)
        for n in xrange(3, int(limit**0.5 + 1.5), 2): 
            if is_prime[n]:
                is_prime[n*n::2*n] = 0
        yield 2
        for i, p in enumerate(is_prime[3::2]):
            if p:
                yield 3 + 2*i 

    @tag('numpy')
    def primes_upto2_gen(limit):
        is_prime = numpy.ones(limit + 1, dtype=numpy.bool)
        for n in xrange(2, int(limit**0.5 + 1.5)): 
            if is_prime[n]:
                is_prime[n*n::n] = 0
        return (i for i in xrange(2, limit+1) if is_prime[i])

    @tag('numpy')
    def primes_upto2_gen2(limit):
        """Generate prime numbers less than limit.

        Assume limit > 2
        """
        yield 2
        is_prime = numpy.ones(limit, dtype=numpy.bool)
        for n in xrange(3, int(limit**.5)+1, 2):
            if is_prime[n]:
                yield n
                is_prime[n*n::n] = False
        for n in xrange(n+2, limit, 2):
            if is_prime[n]:
                yield n

    @tag('numpy')
    def primes_upto2_gen3(limit):
        """Generate prime numbers less than limit.

        Assume limit > 2
        """
        yield 2
        is_prime = numpy.ones(limit, dtype=numpy.bool)
        for n in xrange(3, int(limit**.5)+1, 2):
            if is_prime[n]:
                yield n
                is_prime[n*n::2*n] = False
        for n in xrange(n+2, limit, 2):
            if is_prime[n]:
                yield n

    from numpy import array, bool_, multiply, nonzero, ones, put, resize
    #
    def makepattern(smallprimes):
        pattern = ones(multiply.reduce(smallprimes), dtype=bool_)
        pattern[0] = 0
        for p in smallprimes:
            pattern[p::p] = 0
        return pattern
    #
    @tag('sequence', 'numpy')
    def primes_upto3(limit, smallprimes=(2,3,5,7,11)):    
        sp = array(smallprimes)
        if limit <= sp.max(): return sp[sp <= limit]
        #
        isprime = resize(makepattern(sp), limit + 1) 
        isprime[:2] = 0; put(isprime, sp, 1) 
        #
        for n in range(sp.max() + 2, int(limit**0.5 + 1.5), 2): 
            if isprime[n]:
                isprime[n*n::n] = 0 
        return nonzero(isprime)[0]

    @tag('sequence', 'numpy')
    def primesfrom3to(n):
        """ Returns a array of primes, 3 <= p < n """
        # http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188
        sieve = numpy.ones(n/2, dtype=numpy.bool)
        for i in xrange(3,int(n**0.5)+1,2):
            if sieve[i/2]:
                sieve[i*i/2::i] = False
        return numpy.r_[2,2*numpy.nonzero(sieve)[0][1::]+1]

    @tag('sequence', 'numpy')
    def primesfrom2to(n):
        """ Input n>=6, Returns a array of primes, 2 <= p < n """
        sieve = numpy.ones(n/3+((n+1)&1)*(n%3)/2, dtype=numpy.bool)
        sieve[0] = False
        for i in xrange(int(n**0.5)/3+1):
            if sieve[i]:
                k=3*i+1+(i&1)
                sieve[      ((k*k)/3)      ::2*k] = False
                sieve[(k*k+4*k-2*k*(i&1))/3::2*k] = False
        primes = 3*numpy.nonzero(sieve)[0]+1
        primes += (primes+1)%2
        return numpy.r_[2,3,primes]

    @tag('sequence', 'numpy')
    def primesfrom2to2(n):
        """ Input n>=6, Returns a array of primes, 2 <= p < n """
        sieve = numpy.ones(n/3+((n+1)&1)*(n%3)/2, dtype=numpy.bool)
        sieve[0] = False
        for i in xrange(int(n**0.5)/3+1):
            if sieve[i]:
                k=3*i+1+(i&1)
                sieve[      ((k*k)/3)      ::2*k] = False
                sieve[(k*k+4*k-2*k*(i&1))/3::2*k] = False
        return numpy.r_[2,3,((3*numpy.nonzero(sieve)[0]+1)|1)]

    @tag('sequence', 'numpy')
    def primesfrom2to3(n):
        """ Input n>=6, Returns a array of primes, 2 <= p < n """
        sieve = numpy.ones(n/3 + (n%6==2), dtype=numpy.bool)
        sieve[0] = False
        for i in xrange(int(n**0.5)/3+1):
            if sieve[i]:
                k=3*i+1|1
                sieve[      ((k*k)/3)      ::2*k] = False
                sieve[(k*k+4*k-2*k*(i&1))/3::2*k] = False
        return numpy.r_[2,3,((3*numpy.nonzero(sieve)[0]+1)|1)]
    
    @tag('numpy')
    def primesgen1(n):
        """ Generates all primes < n """
        # http://stackoverflow.com/questions/2897297/speed-up-bitstring-bit-operations-in-python/2908512#2908512
        sieve1 = numpy.ones(n/6+1, dtype=numpy.bool)
        sieve5 = numpy.ones(n/6  , dtype=numpy.bool)
        sieve1[0] = False
        yield 2; yield 3;
        for i in xrange(int(n**0.5)/6+1):
            if sieve1[i]:
                k=6*i+1; yield k;
                sieve1[ ((k*k)/6) ::k] = False
                sieve5[(k*k+4*k)/6::k] = False
            if sieve5[i]:
                k=6*i+5; yield k;
                sieve1[ ((k*k)/6) ::k] = False
                sieve5[(k*k+2*k)/6::k] = False
        for i in xrange(i+1,n/6):
                if sieve1[i]: yield 6*i+1
                if sieve5[i]: yield 6*i+5
        if 1 < n%6 <= 5:
            if sieve1[i+1]: yield 6*i+1

    @tag('numpy')
    def primesgen2(n):
        # http://stackoverflow.com/questions/2897297/speed-up-bitstring-bit-operations-in-python/2908512#2908512
        """ Input n>=30, Generates all primes < n """
        sieve01 = numpy.ones(n/30+1, dtype=numpy.bool)
        sieve07 = numpy.ones(n/30+1, dtype=numpy.bool)
        sieve11 = numpy.ones(n/30+1, dtype=numpy.bool)
        sieve13 = numpy.ones(n/30+1, dtype=numpy.bool)
        sieve17 = numpy.ones(n/30+1, dtype=numpy.bool)
        sieve19 = numpy.ones(n/30+1, dtype=numpy.bool)
        sieve23 = numpy.ones(n/30+1, dtype=numpy.bool)
        sieve29 = numpy.ones(n/30  , dtype=numpy.bool)
        sieve01[0] = False
        yield 2; yield 3; yield 5;
        for i in xrange(int(n**0.5)/30+1):
            if sieve01[i]:
                k=30*i+1; yield k;
                sieve01[     (k*k)/30::k] = False
                sieve07[(k*k+ 6*k)/30::k] = False
                sieve11[(k*k+10*k)/30::k] = False
                sieve13[(k*k+12*k)/30::k] = False
                sieve17[(k*k+16*k)/30::k] = False
                sieve19[(k*k+18*k)/30::k] = False
                sieve23[(k*k+22*k)/30::k] = False
                sieve29[(k*k+28*k)/30::k] = False
            if sieve07[i]:
                k=30*i+7; yield k;
                sieve01[(k*k+ 6*k)/30::k] = False
                sieve07[(k*k+24*k)/30::k] = False
                sieve11[(k*k+16*k)/30::k] = False
                sieve13[(k*k+12*k)/30::k] = False
                sieve17[(k*k+ 4*k)/30::k] = False
                sieve19[     (k*k)/30::k] = False
                sieve23[(k*k+22*k)/30::k] = False
                sieve29[(k*k+10*k)/30::k] = False
            if sieve11[i]:
                k=30*i+11; yield k;
                sieve01[     (k*k)/30::k] = False
                sieve07[(k*k+ 6*k)/30::k] = False
                sieve11[(k*k+20*k)/30::k] = False
                sieve13[(k*k+12*k)/30::k] = False
                sieve17[(k*k+26*k)/30::k] = False
                sieve19[(k*k+18*k)/30::k] = False
                sieve23[(k*k+ 2*k)/30::k] = False
                sieve29[(k*k+ 8*k)/30::k] = False
            if sieve13[i]:
                k=30*i+13; yield k;
                sieve01[(k*k+24*k)/30::k] = False
                sieve07[(k*k+ 6*k)/30::k] = False
                sieve11[(k*k+ 4*k)/30::k] = False
                sieve13[(k*k+18*k)/30::k] = False
                sieve17[(k*k+16*k)/30::k] = False
                sieve19[     (k*k)/30::k] = False
                sieve23[(k*k+28*k)/30::k] = False
                sieve29[(k*k+10*k)/30::k] = False
            if sieve17[i]:
                k=30*i+17; yield k;
                sieve01[(k*k+ 6*k)/30::k] = False
                sieve07[(k*k+24*k)/30::k] = False
                sieve11[(k*k+26*k)/30::k] = False
                sieve13[(k*k+12*k)/30::k] = False
                sieve17[(k*k+14*k)/30::k] = False
                sieve19[     (k*k)/30::k] = False
                sieve23[(k*k+ 2*k)/30::k] = False
                sieve29[(k*k+20*k)/30::k] = False
            if sieve19[i]:
                k=30*i+19; yield k;
                sieve01[     (k*k)/30::k] = False
                sieve07[(k*k+24*k)/30::k] = False
                sieve11[(k*k+10*k)/30::k] = False
                sieve13[(k*k+18*k)/30::k] = False
                sieve17[(k*k+ 4*k)/30::k] = False
                sieve19[(k*k+12*k)/30::k] = False
                sieve23[(k*k+28*k)/30::k] = False
                sieve29[(k*k+22*k)/30::k] = False
            if sieve23[i]:
                k=30*i+23; yield k;
                sieve01[(k*k+24*k)/30::k] = False
                sieve07[(k*k+ 6*k)/30::k] = False
                sieve11[(k*k+14*k)/30::k] = False
                sieve13[(k*k+18*k)/30::k] = False
                sieve17[(k*k+26*k)/30::k] = False
                sieve19[     (k*k)/30::k] = False
                sieve23[(k*k+ 8*k)/30::k] = False
                sieve29[(k*k+20*k)/30::k] = False
            if sieve29[i]:
                k=30*i+29; yield k;
                sieve01[     (k*k)/30::k] = False
                sieve07[(k*k+24*k)/30::k] = False
                sieve11[(k*k+20*k)/30::k] = False
                sieve13[(k*k+18*k)/30::k] = False
                sieve17[(k*k+14*k)/30::k] = False
                sieve19[(k*k+12*k)/30::k] = False
                sieve23[(k*k+ 8*k)/30::k] = False
                sieve29[(k*k+ 2*k)/30::k] = False
        for i in xrange(i+1,n/30):
                if sieve01[i]: yield 30*i+1
                if sieve07[i]: yield 30*i+7
                if sieve11[i]: yield 30*i+11
                if sieve13[i]: yield 30*i+13
                if sieve17[i]: yield 30*i+17
                if sieve19[i]: yield 30*i+19
                if sieve23[i]: yield 30*i+23
                if sieve29[i]: yield 30*i+29
        if n%30 > 1:
            if sieve01[i+1]: yield 30*i+1
        if n%30 > 7:
            if sieve07[i+1]: yield 30*i+7
        if n%30 > 11:
            if sieve11[i+1]: yield 30*i+11
        if n%30 > 13:
            if sieve13[i+1]: yield 30*i+13
        if n%30 > 17:
            if sieve17[i+1]: yield 30*i+17
        if n%30 > 19:
            if sieve19[i+1]: yield 30*i+19
        if n%30 > 23:
            if sieve23[i+1]: yield 30*i+23
            
except ImportError:
    pass


#http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188
@tag('pure', 'sequence')
def primes1(n):
    """ Returns  a list of primes < n """
    sieve = [True] * (n//2)
    for i in range(3,int(n**0.5)+1,2):
        if sieve[i//2]:
            sieve[i*i//2::i] = [False] * ((n-i*i-1)//(2*i)+1)
    return [2] + [2*i+1 for i in range(1,n//2) if sieve[i]]

@tag('pure', 'sequence')
def primes(n):
    """ Returns  a list of primes < n """
    sieve = [True] * n
    for i in range(3,int(n**0.5)+1,2):
        if sieve[i]:
            sieve[i*i::2*i]=[False]*((n-i*i-1)//(2*i)+1)
    return [2] + [i for i in range(3,n,2) if sieve[i]]

try:
#http://stackoverflow.com/questions/2897297/speed-up-bitstring-bit-operations-in-python/2901856#2901856
    import gmpy2

    @tag('gmpy2')
    def prime_numbers(limit=1000000):
        '''Prime number generator. Yields the series
        2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ...
        using Sieve of Eratosthenes.
        '''
        yield 2
        sub_limit = int(limit**0.5)
        # Actual number is 2*bit_position + 1.
        oddnums = gmpy2.xmpz(1)
        current = 0
        while True:
            current += 1
            current = oddnums.bit_scan0(current)
            prime = 2 * current + 1
            if prime > limit:
                break
            yield prime
            # Exclude further multiples of the current prime number
            if prime <= sub_limit:
                for j in range(2*current*(current+1), limit>>1, prime):
                    oddnums.bit_set(j)

    @tag('gmpy2')
    def prime_numbers2(limit=1000000):
        '''Prime number generator. Yields the series
        2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ...
        using Sieve of Eratosthenes.
        '''
        yield 2
        sub_limit = int(limit**0.5)
        # Actual number is 2*bit_position + 1.
        oddnums = gmpy2.xmpz(1)
        f_set = oddnums.bit_set
        f_scan0 = oddnums.bit_scan0
        current = 0
        while True:
            current += 1
            current = f_scan0(current)
            prime = 2 * current + 1
            if prime > limit:
                break
            yield prime
            # Exclude further multiples of the current prime number
            if prime <= sub_limit:
                consume(map(f_set,range(2*current*(current+1), limit>>1, prime)))
                
    @tag('gmpy2')
    def prime_numbers4(limit=1000000):
        '''Prime number generator. Yields the series
        2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ...
        using Sieve of Eratosthenes.
        '''
        sub_limit = int(limit**0.5)
        flags = gmpy2.xmpz(1)
        flags[(limit>>1)+1] = True
        f_scan0 = flags.bit_scan0
        current = 0
        prime = 2
        while prime <= sub_limit:
            yield prime
            current += 1
            current = f_scan0(current)
            prime = 2 * current + 1
            flags[2*current*(current+1):limit>>1:prime] = True
        while prime <= limit:
            yield prime
            current += 1
            current = f_scan0(current)
            prime = 2 * current + 1

#    @tag('gmpy2')
    def gmpy_next_prime_primes(limit):
        next_ = gmpy2.next_prime
        p = next_(1)
        while p < limit:
            yield p
            p = next_(p)        
except ImportError:
    pass


###############################################################################
#http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/2068548#2068548
from math import sqrt, ceil
try:
    import builtins
except ImportError:
    import __builtin__ as builtins
# small primes array
__smallp = ( 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)

@tag('pure', 'sequence')
def sieve_wheel_30(N):
    # http://zerovolt.com/?p=88
    ''' Returns a list of primes <= N using wheel criterion 2*3*5 = 30

Copyright 2009 by zerovolt.com
This code is free for non-commercial purposes, in which case you can just leave this comment as a credit for my work.
If you need this code for commercial purposes, please contact me by sending an email to: info [at] zerovolt [dot] com.'''

    wheel = (2, 3, 5)
    const = 30
    if N < 2:
        return []
    if N <= const:
        pos = 0
        while __smallp[pos] <= N:
            pos += 1
        return list(__smallp[:pos])
    # make the offsets list
    offsets = (7, 11, 13, 17, 19, 23, 29, 1)
    # prepare the list
    p = [2, 3, 5]
    dim = 2 + N // const
    tk1  = [True] * dim
    tk7  = [True] * dim
    tk11 = [True] * dim
    tk13 = [True] * dim
    tk17 = [True] * dim
    tk19 = [True] * dim
    tk23 = [True] * dim
    tk29 = [True] * dim
    tk1[0] = False
    # help dictionary d
    # d[a , b] = c  ==> if I want to find the smallest useful multiple of (30*pos)+a
    # on tkc, then I need the index given by the product of [(30*pos)+a][(30*pos)+b]
    # in general. If b < a, I need [(30*pos)+a][(30*(pos+1))+b]
    d = {}
    for x in offsets:
        for y in offsets:
            res = (x*y) % const
            if res in offsets:
                d[(x, res)] = y
    # another help dictionary: gives tkx calling tmptk[x]
    tmptk = {1:tk1, 7:tk7, 11:tk11, 13:tk13, 17:tk17, 19:tk19, 23:tk23, 29:tk29}
    pos, prime, lastadded, stop = 0, 0, 0, int(ceil(sqrt(N)))
    # inner functions definition
    def del_mult(tk, start, step):
        for k in range(start, len(tk), step):
            tk[k] = False
    # end of inner functions definition
    cpos = const * pos
    while prime < stop:
        # 30k + 7
        if tk7[pos]:
            prime = cpos + 7
            p.append(prime)
            lastadded = 7
            for off in offsets:
                tmp = d[(7, off)]
                start = (pos + prime) if off == 7 else (prime * (const * (pos + 1 if tmp < 7 else 0) + tmp) )//const
                del_mult(tmptk[off], start, prime)
        # 30k + 11
        if tk11[pos]:
            prime = cpos + 11
            p.append(prime)
            lastadded = 11
            for off in offsets:
                tmp = d[(11, off)]
                start = (pos + prime) if off == 11 else (prime * (const * (pos + 1 if tmp < 11 else 0) + tmp) )//const
                del_mult(tmptk[off], start, prime)
        # 30k + 13
        if tk13[pos]:
            prime = cpos + 13
            p.append(prime)
            lastadded = 13
            for off in offsets:
                tmp = d[(13, off)]
                start = (pos + prime) if off == 13 else (prime * (const * (pos + 1 if tmp < 13 else 0) + tmp) )//const
                del_mult(tmptk[off], start, prime)
        # 30k + 17
        if tk17[pos]:
            prime = cpos + 17
            p.append(prime)
            lastadded = 17
            for off in offsets:
                tmp = d[(17, off)]
                start = (pos + prime) if off == 17 else (prime * (const * (pos + 1 if tmp < 17 else 0) + tmp) )//const
                del_mult(tmptk[off], start, prime)
        # 30k + 19
        if tk19[pos]:
            prime = cpos + 19
            p.append(prime)
            lastadded = 19
            for off in offsets:
                tmp = d[(19, off)]
                start = (pos + prime) if off == 19 else (prime * (const * (pos + 1 if tmp < 19 else 0) + tmp) )//const
                del_mult(tmptk[off], start, prime)
        # 30k + 23
        if tk23[pos]:
            prime = cpos + 23
            p.append(prime)
            lastadded = 23
            for off in offsets:
                tmp = d[(23, off)]
                start = (pos + prime) if off == 23 else (prime * (const * (pos + 1 if tmp < 23 else 0) + tmp) )//const
                del_mult(tmptk[off], start, prime)
        # 30k + 29
        if tk29[pos]:
            prime = cpos + 29
            p.append(prime)
            lastadded = 29
            for off in offsets:
                tmp = d[(29, off)]
                start = (pos + prime) if off == 29 else (prime * (const * (pos + 1 if tmp < 29 else 0) + tmp) )//const
                del_mult(tmptk[off], start, prime)
        # now we go back to top tk1, so we need to increase pos by 1
        pos += 1
        cpos = const * pos
        # 30k + 1
        if tk1[pos]:
            prime = cpos + 1
            p.append(prime)
            lastadded = 1
            for off in offsets:
                tmp = d[(1, off)]
                start = (pos + prime) if off == 1 else (prime * (const * pos + tmp) )//const
                del_mult(tmptk[off], start, prime)
    # time to add remaining primes
    # if lastadded == 1, remove last element and start adding them from tk1
    # this way we don't need an "if" within the last while
    if lastadded == 1:
        p.pop()
    # now complete for every other possible prime
    while pos < len(tk1):
        cpos = const * pos
        if tk1[pos]: p.append(cpos + 1)
        if tk7[pos]: p.append(cpos + 7)
        if tk11[pos]: p.append(cpos + 11)
        if tk13[pos]: p.append(cpos + 13)
        if tk17[pos]: p.append(cpos + 17)
        if tk19[pos]: p.append(cpos + 19)
        if tk23[pos]: p.append(cpos + 23)
        if tk29[pos]: p.append(cpos + 29)
        pos += 1
    # remove exceeding if present
    pos = len(p) - 1
    while p[pos] > N:
        pos -= 1
    if pos < len(p) - 1:
        del p[pos+1:]
    # return p list
    return p

@tag('pure', 'sequence')
def sieveOfEratosthenes(n):
    """sieveOfEratosthenes(n): return the list of the primes < n."""
    # Code from: <dickinsm@gmail.com>, Nov 30 2006
    # http://groups.google.com/group/comp.lang.python/msg/f1f10ced88c68c2d
    if n <= 2:
        return []
    sieve = range(3, n, 2)
    if type(sieve) != list:
        sieve = list(sieve)
    top = len(sieve)
    for si in sieve:
        if si:
            bottom = (si*si - 3) // 2
            if bottom >= top:
                break
            sieve[bottom::si] = [0] * -((bottom - top) // si)
    return [2] + [el for el in sieve if el]

@tag('pure', 'sequence')
def sieveOfAtkin(end):
    """sieveOfAtkin(end): return a list of all the prime numbers <end
    using the Sieve of Atkin."""
    # Code by Steve Krenzel, <Sgk284@gmail.com>, improved
    # Code: http://krenzel.info/?p=83
    # Info: http://en.wikipedia.org/wiki/Sieve_of_Atkin
    assert end > 0, "end must be >0"
    lng = ((end-1) // 2)
    sieve = [False] * (lng + 1)

    x_max, x2, xd = int(sqrt((end-1)/4.0)), 0, 4
    for xd in range(4, 8*x_max + 2, 8):
        x2 += xd
        y_max = int(sqrt(end-x2))
        n, n_diff = x2 + y_max*y_max, (y_max << 1) - 1
        if not (n & 1):
            n -= n_diff
            n_diff -= 2
        for d in range((n_diff - 1) << 1, -1, -8):
            m = n % 12
            if m == 1 or m == 5:
                m = n >> 1
                sieve[m] = not sieve[m]
            n -= d

    x_max, x2, xd = int(sqrt((end-1) / 3.0)), 0, 3
    for xd in range(3, 6 * x_max + 2, 6):
        x2 += xd
        y_max = int(sqrt(end-x2))
        n, n_diff = x2 + y_max*y_max, (y_max << 1) - 1
        if not(n & 1):
            n -= n_diff
            n_diff -= 2
        for d in range((n_diff - 1) << 1, -1, -8):
            if n % 12 == 7:
                m = n >> 1
                sieve[m] = not sieve[m]
            n -= d

    x_max, y_min, x2, xd = int((2 + sqrt(4-8*(1-end)))/4), -1, 0, 3
    for x in range(1, x_max + 1):
        x2 += xd
        xd += 6
        if x2 >= end: y_min = (((int(ceil(sqrt(x2 - end))) - 1) << 1) - 2) << 1
        n, n_diff = ((x*x + x) << 1) - 1, (((x-1) << 1) - 2) << 1
        for d in range(n_diff, y_min, -8):
            if n % 12 == 11:
                m = n >> 1
                sieve[m] = not sieve[m]
            n += d

    primes = [2, 3]
    if end <= 3:
        return primes[:max(0,end-2)]

    for n in range(5 >> 1, (int(sqrt(end))+1) >> 1):
        if sieve[n]:
            primes.append((n << 1) + 1)
            aux = (n << 1) + 1
            aux *= aux
            for k in range(aux, end, 2 * aux):
                sieve[k >> 1] = False

    s  = int(sqrt(end)) + 1
    if s  % 2 == 0:
        s += 1
    primes.extend([i for i in range(s, end, 2) if sieve[i >> 1]])

    return primes
try:
    import numpy as np

    @tag('sequence', 'numpy')
    def ambi_sieve(n):
        # http://tommih.blogspot.com/2009/04/fast-prime-number-generator.html
        s = np.arange(3, n, 2)
        for m in builtins.xrange(3, int(n ** 0.5)+1, 2): 
            if s[(m-3)/2]: 
                s[(m*m-3)/2::m]=0
        return np.r_[2, s[s>0]]
except ImportError:
    pass

@tag('pure', 'sequence')
def ambi_sieve_plain(n):
    s = list(range(3, n, 2))
    for m in range(3, int(n**0.5)+1, 2): 
        if s[(m-3)//2]: 
            for t in range((m*m-3)//2,(n>>1)-1,m):
                s[t]=0
    return [2]+[t for t in s if t>0]

@tag('pure', 'sequence')
def sundaram3(max_n):
    numbers = list(range(3, max_n+1, 2))
    half = (max_n)//2
    initial = 4

    for step in range(3, max_n+1, 2):
        for i in range(initial, half, step):
            numbers[i-1] = 0
        initial += 2*(step+1)

        if initial > half:
            return [2] + [n for n in numbers if n]

###############################################################################

@tag('pure', 'sequence')
def rwh_primes2(n):
    # http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188
    """ Input n>=6, Returns a list of primes, 2 <= p < n """
    n, correction = n-n%6+6, 2-(n%6>1)
    sieve = [True] * (n//3)
    for i in range(1,int(n**0.5)//3+1):
      if sieve[i]:
        k=3*i+1|1
        sieve[      k*k//3      ::2*k] = [False] * ((n//6-k*k//6-1)//k+1)
        sieve[k*(k-2*(i&1)+4)//3::2*k] = [False] * ((n//6-k*(k-2*(i&1)+4)//6-1)//k+1)
    return [2,3] + [3*i+1|1 for i in range(1,n//3-correction) if sieve[i]]

@tag('pure', 'sequence')
def sieveOfErat_Noldorin(end):
  #  http://stackoverflow.com/questions/1023768/sieve-of-atkin-explanation/1023777#1023777
  if end < 2: return []  

  #The array doesn't need to include even numbers  
  lng = ((end//2)-1+end%2)  

  # Create array and assume all numbers in array are prime  
  sieve = [True]*(lng+1)  

  # In the following code, you're going to see some funky  
  # bit shifting and stuff, this is just transforming i and j  
  # so that they represent the proper elements in the array.  
  # The transforming is not optimal, and the number of  
  # operations involved can be reduced.  

  # Only go up to square root of the end  
  for i in range(int(sqrt(end)) >> 1):  

    # Skip numbers that aren't marked as prime  
    if not sieve[i]: continue  

    # Unmark all multiples of i, starting at i**2  
    for j in range( (i*(i + 3) << 1) + 3, lng, (i << 1) + 3):  
      sieve[j] = False  

  # Don't forget 2!  
  primes = [2]  

  # Gather all the primes into a list, leaving out the composite numbers  
  primes.extend([(i << 1) + 3 for i in range(lng) if sieve[i]])  

  return primes
        

try:
    import primegen # http://cr.yp.to/primegen.html

    @tag('primegen')
    def primegen_upto(n):
        pg = primegen.primegen()
        primegen.primegen_init(pg)
        next_ = primegen.primegen_next
        p = next_(pg)
        while p < n:
            yield p
            p = next_(pg)
except ImportError:
    pass

@tag('pure')
def eratosthenes_dugres(n):
    # http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3328618#3328618
    N=list(range(n+1))
    z=[0]*(n//2)   
    for i in range(2, int(n**.5)+1):
        if N[i]:
            N[i*i::i] = z[:(n//i)-i+1]
    return filter(None, N[2:])

try:
    bytearray # `bytearray` is not present in pypy
    
    @tag('pure')
    def prime_pure(limit=1000000):
    # http://stackoverflow.com/questions/2897297/speed-up-bitstring-bit-operations-in-python/2897459#2897459    
        yield 2
        flags = bytearray((limit + 7) // 8)
        sub_limit = int(limit**0.5)
        for i in range(3, limit, 2):
            byte, bit = divmod(i, 8)
            if not flags[byte] & (128 >> bit):
                yield i
                if i <= sub_limit:
                    for j in range(i*3, limit, i*2):
                        byte, bit = divmod(j, 8)
                        flags[byte] |= (128 >> bit)
except NameError:
    pass
#
################################################################################

def consume(iterator):
    collections.deque(iterator, maxlen=0)

@profile
def test(n=1000000):
    nsmall = 10000
    result = list(register.functions[0](nsmall))
    for f in register.functions[1:]:
        for expected, got in zip(result, f(nsmall)):
            assert expected == got, (expected, got, f.__name__)

    results = [test_func(f, n, quiet=True) for f in register.functions]
    results.sort(key=lambda x:x[1])
    for r in results:
        print25('%s:%s %s' % (
            str(r[0]).ljust(25), ('%d ms' % (int(r[1]*1e3+0.5),)).rjust(10), r[2]))

def test_func(f, n=1000000, quiet=False, repeat=3, ntimes=10):
    stmt = '%s(%d)' % (f.__name__, n)
    setup = 'from %s import %s' % (__name__, f.__name__)
    if not 'sequence' in f.tags:
        stmt = 'consume(%s)' % (stmt,)
        setup += '; from %s import consume' % (__name__,)
        
    t = timeit.Timer(stmt, setup)
    try:
        results = [r/ntimes for r in t.repeat(repeat=repeat, number=ntimes)]
    except Exception:
        time = -1
    else:
        time = min(results)
    result = (f.__name__, time, ':'.join(f.tags))
    if not quiet:
        print25(*result)
    return result

if __name__=="__main__":    
    test()
    try:
        for f in [primesgen2, primesfrom2to3]:
            test_func(f, n=10**9, repeat=1, ntimes=1)
    except NameError:
        pass
    
# Python2.6 n=1e6
# primesfrom2to3           :      7 ms numpy:sequence
# primesfrom2to2           :      7 ms numpy:sequence
# ambi_sieve               :      9 ms numpy:sequence
# primesfrom2to            :      9 ms numpy:sequence
# primesfrom3to            :      9 ms numpy:sequence
# nolfonzo_prime6          :     12 ms numpy:sequence
# primes_upto4             :     15 ms numpy:sequence
# primes_upto3             :     15 ms numpy:sequence
# primes_upto2             :     17 ms numpy:sequence
# primegen_upto            :     31 ms primegen
# prime_numbers4           :     32 ms gmpy2
# primesgen2               :     35 ms numpy
# primesgen1               :     38 ms numpy
# rwh_primes2              :     44 ms pure:sequence
# primes_upto2_gen3        :     47 ms numpy
# primes_upto2_gen2        :     48 ms numpy
# primes1                  :     49 ms pure:sequence
# primes                   :     50 ms pure:sequence
# primes_upto4_gen         :     53 ms numpy
# sieve_wheel_30           :     56 ms pure:sequence
# sieveOfEratosthenes      :     79 ms pure:sequence
# prime_numbers2           :     83 ms gmpy2
# primes_upto2_gen         :     91 ms numpy
# sieveOfErat_Noldorin     :    112 ms pure:sequence
# ambi_sieve_plain         :    145 ms pure:sequence
# prime_numbers            :    146 ms gmpy2
# eratosthenes_dugres      :    161 ms pure
# sieveOfAtkin             :    219 ms pure:sequence
# sundaram3                :    235 ms pure:sequence
# iprimes_upto             :    281 ms pure
# iprimes_upto_enumerate   :    283 ms pure
# prime_pure               :    605 ms pure

# Python2.6 n=1e9
# primesgen1 35.7735440731 numpy
# primesgen2 27.4978778362 numpy
# primesfrom2to3 9.91236019135 numpy:sequence

# Python3 n=1e6
# prime_numbers4           :     32 ms gmpy2
# rwh_primes2              :     38 ms pure:sequence
# sieve_wheel_30           :     44 ms pure:sequence
# primes                   :     48 ms pure:sequence
# primes1                  :     49 ms pure:sequence
# sieveOfEratosthenes      :     49 ms pure:sequence
# prime_numbers2           :     92 ms gmpy2
# sieveOfErat_Noldorin     :     93 ms pure:sequence
# ambi_sieve_plain         :    109 ms pure:sequence
# prime_numbers            :    112 ms gmpy2
# eratosthenes_dugres      :    112 ms pure
# sundaram3                :    187 ms pure:sequence
# sieveOfAtkin             :    199 ms pure:sequence
# iprimes_upto_enumerate   :    225 ms pure
# iprimes_upto             :    227 ms pure
# prime_pure               :    500 ms pure

# pypy n=1e6
# rwh_primes2              :     84 ms pure:sequence
# sieve_wheel_30           :    113 ms pure:sequence
# primes                   :    115 ms pure:sequence
# primes1                  :    116 ms pure:sequence
# sieveOfEratosthenes      :    135 ms pure:sequence
# sieveOfErat_Noldorin     :    251 ms pure:sequence
# ambi_sieve_plain         :    252 ms pure:sequence
# sieveOfAtkin             :    347 ms pure:sequence
# sundaram3                :    436 ms pure:sequence

