def alexandrian(n):
R = [r**2 + 1 for r in range(n + 1)]
P = eulertools.primesieve(n)
R = [[] for r in range(n + 1)]
for p in P:
soln = eulertools.modsqrt_prime(-1, p)
for s in soln:
for k in xrange(s, n + 1, p):
r2 = k**2 + 1
R[k].append(p)
F = []
for r, V in enumerate(R):
r2 = r**2 + 1
F.append(factors(r2, V))
output = set()
for r, X in enumerate(F):
if r == 0:
continue
r2 = r**2 + 1
for x in X:
p = x + r
q = (r2 / x) + r
A = abs(r * p * q)
if p * q - p * r - q * r != 1:
raise
output.add(A)
return sorted(list(output))
def test():
P = eulertools.primesieve(10000)
sol = 0
for x in range(2, 10001):
if d(P, d(P, x)) == x and x != d(P, x):
sol += x
return sol
def test(): P = eulertools.primesieve(10000) sol = 0 for x in range(2,10001): if d(P,d(P,x))==x and x!= d(P,x): sol +=x return sol
def test():
P = set(eulertools.primesieve(10000))
max_counter = 0
for a in range(-100, 1000):
for b in range(-100, 1000):
x = 0
while True:
f = x**2 + a * x + b
if f not in P:
break
x += 1
if x > max_counter:
max_counter = x
#print x,a,b,a*b
sol = a * b
return sol
from eulertools import primesieve N = 10**9 Q = 100 P = primesieve(Q) prev = set([1]) solution = [prev] for i in range(1,100): new = set([a*p for p in P for a in prev if a*p <= N]) print i,len(new) if len(new)==0: break solution.append(new) prev = new solution = set.union(*solution) print len(solution)
for i in range(n):
if i == 0:
Q = (1, 2, 3, 4, 5, 6, 7, 8, 9)
else:
Q = (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
for q in Q:
sy = list(sx)
sy[i] = str(q)
y = int(''.join(sy))
if eulertools.is_prime(y):
return False
return True
P = eulertools.primesieve(10**6)
output = []
qMax = 10**12
for p1 in P:
for p2 in P:
if p1 != p2:
q = (p1**2) * (p2**3)
if q > qMax:
break
output.append((q, p1, p2))
output.sort()
print len(output)
from eulertools import primesieve from math import factorial def nCr(n,r): return factorial(n)/(factorial(r)*factorial(n-r)) def containsFactor(x,F): for f in F: if x % f ==0: return True return False N = 51 P = primesieve(N) F = [p**2 for p in P] output = set([nCr(n,r) for n in range(0,N) for r in range(0,n+1)]) output = [a for a in output if not containsFactor(a,F)] print output print len(output) print sum(output)
import eulertools N = 10**8 P = eulertools.primesieve(N) counter = 0 print 'Go' for p1 in P: for p2 in P: if p2 > p1: break q = p1*p2 if q > N: break counter +=1 print counter
import eulertools
import numpy
nmax = 50 * 10**6
A = [0 for i in xrange(nmax + 1)]
P = set(eulertools.primesieve(int(2**0.5 * nmax) + 1))
print len(P)
print 'Done'
for p in sorted(list(P)):
if p > 2:
roots = eulertools.modsqrt_prime((p + 1) / 2, p)
if len(roots) > 0:
for r in roots:
if 2 * r * r - 1 in P:
a = r + p
else:
a = r
for s in xrange(a, nmax + 1, p):
A[s] = 1
print nmax - sum(A) - 1
def test():
x = 600851475143
rootx = int(x**0.5) + 1
P = eulertools.primesieve(rootx)
factors = [p for p in P if x % p == 0]
return max(factors)
def test(): P = eulertools.primesieve(100) solution = shortest_length(P,47547) return solution
def test(): return sum(eulertools.primesieve(2000000))
from eulertools import primesieve
N = 10**9
Q = 100
P = primesieve(Q)
prev = set([1])
solution = [prev]
for i in range(1, 100):
new = set([a * p for p in P for a in prev if a * p <= N])
print i, len(new)
if len(new) == 0:
break
solution.append(new)
prev = new
solution = set.union(*solution)
print len(solution)
def test():
n = 10001
max_size = int(n * math.log(n * math.log(n)))
A = eulertools.primesieve(max_size)
return A[n - 1]
from itertools import combinations
from eulertools import gcd, factorise, primesieve, product
r = 12017639147
#r = 1000001
h = (r + 3) / 2
P = primesieve(int(h**0.5) + 1)
F = set(factorise(P, h))
N = len(F)
total = int(h / 3)
print h, N, F
for k in range(1, N + 1):
for X in combinations(F, k):
prod = product(X)
term = int(2 * h / (3 * prod)) - int(h / (3 * prod))
sign = (-1)**k
print X, k, sign, prod
total += sign * term
print total
#print 80840
def test():
x = 600851475143
rootx = int(x**0.5)+1
P = eulertools.primesieve(rootx)
factors = [p for p in P if x% p ==0]
return max(factors)
import eulertools
N = 10**8
P = eulertools.primesieve(N)
counter = 0
print 'Go'
for p1 in P:
for p2 in P:
if p2 > p1:
break
q = p1 * p2
if q > N:
break
counter += 1
print counter
import eulertools import numpy nmax = 50*10**6 A = [0 for i in xrange(nmax+1)] P = set(eulertools.primesieve(int(2**0.5*nmax)+1)) print len(P) print 'Done' for p in sorted(list(P)): if p>2: roots = eulertools.modsqrt_prime((p+1)/2,p) if len(roots)>0: for r in roots: if 2*r*r-1 in P: a = r+p else: a = r for s in xrange(a,nmax+1,p): A[s]=1 print nmax-sum(A)-1
