def problem60():
"""Problem 60 - Prime pair sets"""
sieve.extend(
PRIME_LIMIT) # find_prime_concatenates requires non-empty sieve
solutions = []
find_prime_concatenates([], 5, PRIME_LIMIT, solutions)
return min(sum(s) for s in solutions)
def test_generate():
from sympy.ntheory.generate import sieve
sieve._reset()
assert nextprime(-4) == 2
assert nextprime(2) == 3
assert nextprime(5) == 7
assert nextprime(12) == 13
assert prevprime(3) == 2
assert prevprime(7) == 5
assert prevprime(13) == 11
assert prevprime(19) == 17
assert prevprime(20) == 19
sieve.extend_to_no(9)
assert sieve._list[-1] == 23
assert sieve._list[-1] < 31
assert 31 in sieve
assert nextprime(90) == 97
assert nextprime(10**40) == (10**40 + 121)
assert prevprime(97) == 89
assert prevprime(10**40) == (10**40 - 17)
assert list(sieve.primerange(10, 1)) == []
assert list(primerange(10, 1)) == []
assert list(primerange(2, 7)) == [2, 3, 5]
assert list(primerange(2, 10)) == [2, 3, 5, 7]
assert list(primerange(1050, 1100)) == [1051, 1061,
1063, 1069, 1087, 1091, 1093, 1097]
s = Sieve()
for i in range(30, 2350, 376):
for j in range(2, 5096, 1139):
A = list(s.primerange(i, i + j))
B = list(primerange(i, i + j))
assert A == B
s = Sieve()
assert s[10] == 29
assert nextprime(2, 2) == 5
raises(ValueError, lambda: totient(0))
raises(ValueError, lambda: reduced_totient(0))
raises(ValueError, lambda: primorial(0))
assert mr(1, [2]) is False
func = lambda i: (i**2 + 1) % 51
assert next(cycle_length(func, 4)) == (6, 2)
assert list(cycle_length(func, 4, values=True)) == \
[17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14]
assert next(cycle_length(func, 4, nmax=5)) == (5, None)
assert list(cycle_length(func, 4, nmax=5, values=True)) == \
[17, 35, 2, 5, 26]
sieve.extend(3000)
assert nextprime(2968) == 2969
assert prevprime(2930) == 2927
raises(ValueError, lambda: prevprime(1))
def test_prime():
assert prime(1) == 2
assert prime(2) == 3
assert prime(5) == 11
assert prime(11) == 31
assert prime(57) == 269
assert prime(296) == 1949
assert prime(559) == 4051
assert prime(3000) == 27449
assert prime(4096) == 38873
assert prime(9096) == 94321
assert prime(25023) == 287341
raises(ValueError, lambda: prime(0))
sieve.extend(3000)
assert prime(401) == 2749
def test_prime():
assert prime(1) == 2
assert prime(2) == 3
assert prime(5) == 11
assert prime(11) == 31
assert prime(57) == 269
assert prime(296) == 1949
assert prime(559) == 4051
assert prime(3000) == 27449
assert prime(4096) == 38873
assert prime(9096) == 94321
assert prime(25023) == 287341
raises(ValueError, lambda: prime(0))
sieve.extend(3000)
assert prime(401) == 2749
def problem12(nr_divisors):
"""Problem 12 - Highly divisible triangular number"""
sieve.extend(MAX_PRIME)
x = 1
while True:
nr = number_of_devisors(x)
if x % 2 == 0: # even
nr = number_of_devisors(x // 2) * number_of_devisors(x + 1)
else: # uneven
nr = number_of_devisors(x) * number_of_devisors((x + 1) // 2)
if nr >= nr_divisors:
break
x += 1
return x * (x + 1) // 2
def test_compositepi():
assert compositepi(1) == 0
assert compositepi(2) == 0
assert compositepi(5) == 1
assert compositepi(11) == 5
assert compositepi(57) == 40
assert compositepi(296) == 233
assert compositepi(559) == 456
assert compositepi(3000) == 2569
assert compositepi(4096) == 3531
assert compositepi(9096) == 7967
assert compositepi(25023) == 22259
assert compositepi(10**8) == 94238544
assert compositepi(253425253) == 239568856
assert compositepi(8769575643) == 8368111320
sieve.extend(3000)
assert compositepi(2321) == 1976
def test_primepi():
assert primepi(1) == 0
assert primepi(2) == 1
assert primepi(5) == 3
assert primepi(11) == 5
assert primepi(57) == 16
assert primepi(296) == 62
assert primepi(559) == 102
assert primepi(3000) == 430
assert primepi(4096) == 564
assert primepi(9096) == 1128
assert primepi(25023) == 2763
assert primepi(10**8) == 5761455
assert primepi(253425253) == 13856396
assert primepi(8769575643) == 401464322
sieve.extend(3000)
assert primepi(2000) == 303
def test_primepi():
assert primepi(1) == 0
assert primepi(2) == 1
assert primepi(5) == 3
assert primepi(11) == 5
assert primepi(57) == 16
assert primepi(296) == 62
assert primepi(559) == 102
assert primepi(3000) == 430
assert primepi(4096) == 564
assert primepi(9096) == 1128
assert primepi(25023) == 2763
assert primepi(10**8) == 5761455
assert primepi(253425253) == 13856396
assert primepi(8769575643) == 401464322
sieve.extend(3000)
assert primepi(2000) == 303
def test_compositepi():
assert compositepi(1) == 0
assert compositepi(2) == 0
assert compositepi(5) == 1
assert compositepi(11) == 5
assert compositepi(57) == 40
assert compositepi(296) == 233
assert compositepi(559) == 456
assert compositepi(3000) == 2569
assert compositepi(4096) == 3531
assert compositepi(9096) == 7967
assert compositepi(25023) == 22259
assert compositepi(10**8) == 94238544
assert compositepi(253425253) == 239568856
assert compositepi(8769575643) == 8368111320
sieve.extend(3000)
assert compositepi(2321) == 1976
def test_prime():
assert prime(1) == 2
assert prime(2) == 3
assert prime(5) == 11
assert prime(11) == 31
assert prime(57) == 269
assert prime(296) == 1949
assert prime(559) == 4051
assert prime(3000) == 27449
assert prime(4096) == 38873
assert prime(9096) == 94321
assert prime(25023) == 287341
assert prime(10000000) == 179424673 # issue #20951
assert prime(99999999) == 2038074739
raises(ValueError, lambda: prime(0))
sieve.extend(3000)
assert prime(401) == 2749
raises(ValueError, lambda: prime(-1))
def test_composite():
from sympy.ntheory.generate import sieve
sieve._reset()
assert composite(1) == 4
assert composite(2) == 6
assert composite(5) == 10
assert composite(11) == 20
assert composite(41) == 58
assert composite(57) == 80
assert composite(296) == 370
assert composite(559) == 684
assert composite(3000) == 3488
assert composite(4096) == 4736
assert composite(9096) == 10368
assert composite(25023) == 28088
sieve.extend(3000)
assert composite(1957) == 2300
assert composite(2568) == 2998
raises(ValueError, lambda: composite(0))
def test_composite():
from sympy.ntheory.generate import sieve
sieve._reset()
assert composite(1) == 4
assert composite(2) == 6
assert composite(5) == 10
assert composite(11) == 20
assert composite(41) == 58
assert composite(57) == 80
assert composite(296) == 370
assert composite(559) == 684
assert composite(3000) == 3488
assert composite(4096) == 4736
assert composite(9096) == 10368
assert composite(25023) == 28088
sieve.extend(3000)
assert composite(1957) == 2300
assert composite(2568) == 2998
raises(ValueError, lambda: composite(0))
def main():
n = 1000000
output = 0
sieve.extend(1000000)
for i in range(2, n + 1):
for j in range(1, i):
if (i in sieve or j in sieve):
output += 1
elif gcd(i, j) == 1:
output += 1
print(i)
print("output: " + str(output))
# output = n - 1 # all numbers have 1/n apart from 1
# primes = 1 # primes less than 3
# for i in range(3,n + 1):
# if isprime(i): # if i is prime we add i - 2 to the output as we've already counted 1/i and don't count i/i
# primes += 1 # there's one more prime < i
# output += i - 2 # see above
# else: # number is not prime
# output += primes - len(primefactors(i))
# print(i)
print("Count: " + str(output))
def test_primepi():
assert primepi(-1) == 0
assert primepi(1) == 0
assert primepi(2) == 1
assert primepi(Rational(7, 2)) == 2
assert primepi(3.5) == 2
assert primepi(5) == 3
assert primepi(11) == 5
assert primepi(57) == 16
assert primepi(296) == 62
assert primepi(559) == 102
assert primepi(3000) == 430
assert primepi(4096) == 564
assert primepi(9096) == 1128
assert primepi(25023) == 2763
assert primepi(10**8) == 5761455
assert primepi(253425253) == 13856396
assert primepi(8769575643) == 401464322
sieve.extend(3000)
assert primepi(2000) == 303
n = Symbol('n')
assert primepi(n).subs(n, 2) == 1
r = Symbol('r', real=True)
assert primepi(r).subs(r, 2) == 1
assert primepi(S.Infinity) is S.Infinity
assert primepi(S.NegativeInfinity) == 0
assert limit(primepi(n), n, 100) == 25
raises(ValueError, lambda: primepi(I))
raises(ValueError, lambda: primepi(1 + I))
raises(ValueError, lambda: primepi(zoo))
raises(ValueError, lambda: primepi(nan))
from sympy import sieve, isprime
from numpy import sqrt
sieve.extend(5)
current = 9
conjecture = True
while conjecture:
while isprime(current):
current += 2
sieve.extend(current)
isGood = False
for i in sieve._list:
if sqrt((current - i) / 2) % 1 == 0:
isGood = True
break
if not isGood:
print(current)
conjecture = False
current += 2
print("Done")
# Answer: 5777
def test_generate():
from sympy.ntheory.generate import sieve
sieve._reset()
assert nextprime(-4) == 2
assert nextprime(2) == 3
assert nextprime(5) == 7
assert nextprime(12) == 13
assert prevprime(3) == 2
assert prevprime(7) == 5
assert prevprime(13) == 11
assert prevprime(19) == 17
assert prevprime(20) == 19
sieve.extend_to_no(9)
assert sieve._list[-1] == 23
assert sieve._list[-1] < 31
assert 31 in sieve
assert nextprime(90) == 97
assert nextprime(10**40) == (10**40 + 121)
assert prevprime(97) == 89
assert prevprime(10**40) == (10**40 - 17)
assert list(sieve.primerange(10, 1)) == []
assert list(sieve.primerange(5, 9)) == [5, 7]
sieve._reset(prime=True)
assert list(sieve.primerange(2, 12)) == [2, 3, 5, 7, 11]
assert list(sieve.totientrange(5, 15)) == [4, 2, 6, 4, 6, 4, 10, 4, 12, 6]
sieve._reset(totient=True)
assert list(sieve.totientrange(3, 13)) == [2, 2, 4, 2, 6, 4, 6, 4, 10, 4]
assert list(sieve.totientrange(900, 1000)) == [totient(x) for x in range(900, 1000)]
assert list(sieve.totientrange(0, 1)) == []
assert list(sieve.totientrange(1, 2)) == [1]
assert list(sieve.mobiusrange(5, 15)) == [-1, 1, -1, 0, 0, 1, -1, 0, -1, 1]
sieve._reset(mobius=True)
assert list(sieve.mobiusrange(3, 13)) == [-1, 0, -1, 1, -1, 0, 0, 1, -1, 0]
assert list(sieve.mobiusrange(1050, 1100)) == [mobius(x) for x in range(1050, 1100)]
assert list(sieve.mobiusrange(0, 1)) == []
assert list(sieve.mobiusrange(1, 2)) == [1]
assert list(primerange(10, 1)) == []
assert list(primerange(2, 7)) == [2, 3, 5]
assert list(primerange(2, 10)) == [2, 3, 5, 7]
assert list(primerange(1050, 1100)) == [1051, 1061,
1063, 1069, 1087, 1091, 1093, 1097]
s = Sieve()
for i in range(30, 2350, 376):
for j in range(2, 5096, 1139):
A = list(s.primerange(i, i + j))
B = list(primerange(i, i + j))
assert A == B
s = Sieve()
assert s[10] == 29
assert nextprime(2, 2) == 5
raises(ValueError, lambda: totient(0))
raises(ValueError, lambda: reduced_totient(0))
raises(ValueError, lambda: primorial(0))
assert mr(1, [2]) is False
func = lambda i: (i**2 + 1) % 51
assert next(cycle_length(func, 4)) == (6, 2)
assert list(cycle_length(func, 4, values=True)) == \
[17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14]
assert next(cycle_length(func, 4, nmax=5)) == (5, None)
assert list(cycle_length(func, 4, nmax=5, values=True)) == \
[17, 35, 2, 5, 26]
sieve.extend(3000)
assert nextprime(2968) == 2969
assert prevprime(2930) == 2927
raises(ValueError, lambda: prevprime(1))
sqrtlim = int(sqrt(lim))
gen = set()
S = 1
def nextgen():
global gen,S
#eprint(gen)
newgen = set()
S += len(gen)
for n in gen:
for p in sieve.primerange(1,min(sqrtlim,lim//n+1)):
newgen.add(p*n)
del gen
gen = newgen
if __name__=="__main__":
tic()
sieve.extend(sqrtlim)
for p1 in sieve.primerange(1,sqrtlim):
for p2 in sieve.primerange(p1,min(sqrtlim,lim//p1+1)):
if p1*p2 > lim:
break
for p3 in sieve.primerange(p2,min(sqrtlim,lim//(p1*p2)+1,p1*p2+1)):
gen.add(p1*p2*p3)
while len(gen)>0:
nextgen()
print(S)
toc()
exit()
import math import itertools from sympy import sieve length = 100000 sieve.extend(length) divisors = [0]*length divisors[1] = [1] divisors[2] = [1,2] def get_two_factors(n): i = 2 while((n%i)!=0): i+=1 return (n/i, i) for i in range(3,length): if i in sieve: divisors[i] = [1,i] else: a,b = get_two_factors(i) divisors[i] = list(set([x * y for (x, y) in itertools.product(divisors[a], divisors[b])])) for i in range(1,length): small = i if i%2!=0 else i/2 big = i+1 if (i+1)%2!=0 else (i+1)/2 total = len(divisors[small]) * len(divisors[big]) if (total)>500: print small,big,total break
from sympy import sieve # library for primes
def get_pfct(n):
i = 2;
factors = []
while i * i <= n:
if n % i:
i += 1
else:
n //= i
factors.append(i)
if n > 1:
factors.append(n)
return len(factors)
sieve.extend(110) # first 110 primes...
primes = sieve._list
pool = []
for each in range(0, 121):
pool.append(get_pfct(each))
for i, each in enumerate(pool):
if each in primes:
print(i, end=',')
ezpz
"""
from Euler.tictoc import *
from sympy import sieve
from Euler.digits import intToDigits, digitsToInt
def rotations(n):
dig = intToDigits(n)
for i in range(1, len(dig)):
yield digitsToInt(dig[i:] + dig[:i])
if __name__ == "__main__":
tic()
sieve.extend(1e6)
primes = {p for p in sieve._list if p < 1e6}
found = set()
for p in primes:
if p in found:
continue
rot = rotations(p)
if all(map(lambda r: r in primes, rot)):
found.add(p)
for r in rot:
found.add(r)
print(len(found))
toc()
exit()
def primeRange2(numRange):
sieve.extend(numRange)
print("\nHere are the prime numbers up to the number you entered: ", sieve._list)
print("\nThere are", len(sieve._list), "prime numbers up to", numRange)
#NumberTh functions with SymPy package
#-------------------------------------
from sympy import ntheory
from sympy import sieve
from sympy import generate
sieve._reset(
) # An infinite list of prime numbers, implemented as a dynamically growing sieve of Eratosthenes.#When a lookup is
# requested involving an odd number that has not been sieved,the sieve is automatically extended up to that number.
print(13 in sieve) # prints 'True' if the input is prime,
sieve.extend(40) #for input n grows the sieve to contain all primes <= n
print(sieve[10]) # returns n th prime
sieve.extend_to_no(15) # extends upto n th prime
# mobiusrange(a,b) genarates Mobius numbers for the range [a,b), means output will be a list.
Mob_func = sieve.mobiusrange(7, 12)
print('\n Mobius function outputs in the given range:', [i for i in Mob_func])
# primerange(a,b) generates all prime numbers in the range [a,b)
Prime_list = sieve.primerange(1, 5)
print('\n Primes in the given range are :', [i for i in Prime_list])
#search(n) return the indices i, j of the primes that bound n. If n is prime then i == j. Although n can be an expression,
# if ceiling cannot convert it to an integer then an error will be raised.
x, y = sieve.search(25)
print('\n The given input is in between ', '(', x, ',', y, ')', 'th Primes \n')
from PermuteString import PermuteString
from sympy import sieve
import itertools
sieve.extend(1000000)
primes = sieve._list
stringprimes=[]
for prime in primes:
stringprimes.append(str(prime))
count = 0
primeset = set(stringprimes)
excluded_digits = set(['0','2','4','5','6','8'])
for p in stringprimes:
if not excluded_digits.isdisjoint(set(p)):
continue
perm = PermuteString(p)
if set(perm) <= primeset:
count += 1
print p
count += 2
print count
from sympy import sieve
sieve.extend(2000000)
maxcount = 0
amax = 0
bmax = 0
for b in sieve.primerange(0,1000):
for a in xrange(-2*(b/2)-1, 1000, 2):
term = lambda n: n**2 + a*n + b
i = 0
count = 0
while sieve.__contains__(term(i)):
i += 1
count += 1
if count > maxcount:
maxcount = count
amax = a
bmax = b
print maxcount
print amax, bmax
from sympy import isprime, divisors, sieve
sieve.extend(100000000)
def cond(n):
if not isprime(int(2 + n/2)): return False
else:
d = divisors(n)
for i in range(2,int(len(d)/2)):
if not isprime(d[i] + int(n/d[i])): return False
return True
U = [i-1 for i in sieve.primerange(1,100000000) if (i-1) % 4 != 0]
print(sum([n for n in U if cond(n)]))
