def main():
lst = prepare_lst()
result = {10}
for a in sorted(lst):
for addition in ADDITIONS:
if not euler.miller_rabin(a * a + addition):
break
else:
for addition in (19, 21):
if euler.miller_rabin(a * a + addition):
break
else:
result.add(a)
return sum(result)
def main():
total = 0
i = 0
for n in itertools.count(4):
if (n & 1 and n % 5 and not euler.miller_rabin(n)
and not (n - 1) % find(n)):
i += 1
total += n
if i == LIMIT:
break
return total
def main():
prime_list = euler.prime_list(10**6)
masks = collections.defaultdict(int)
for prime in prime_list:
for mask in all_masks(prime):
masks[mask] += 1
if masks[mask] == 8:
start = 1 if mask.startswith('*') else 0
for i in range(start, 10):
n = int(mask.replace('*', str(i)))
if euler.miller_rabin(n):
return n
return 0
def main():
diff = 2
primes = 3
total = 5
ratio = primes / total
temp = 9
while ratio >= 0.1:
diff += 2
for _ in range(3):
temp += diff
if euler.miller_rabin(temp):
primes += 1
temp += diff
total += 4
ratio = primes / total
return diff + 1
def main():
last = 0
for i in range(100000):
s = str(i * 56789)
if s.endswith('99999'):
last = i
last_digit_sum = sum((9 * i) % 10 for i in range(10))
for i in range(
int(round(1 / 0.00000000138, -5)) + last,
int(round(1 / 0.00000000137, -5)),
10 ** 5
):
if euler.miller_rabin(i) and euler.prime_cycle_length(i) == i - 1:
n = i - 1
result = last_digit_sum * (n // 10)
for j in range(n % 10):
result += (9 * j) % 10
return result
return 0
def main():
total = 0
lst = list(range(LIMIT1))
lst[1] = 0
nums = [2 * n * n - 1 for n in range(1000)]
for prime in euler.prime_list(1000):
for rest in range(1, prime):
num = nums[rest]
if not num % prime:
if num == prime:
total += 1
for i in range(rest, LIMIT1, prime):
lst[i] = 0
for n in lst:
if n:
num = 2 * n * n - 1
if euler.miller_rabin(num):
total += 1
for i in range(n, LIMIT1, num):
lst[i] = 0
return total
def s_func(digit):
for m in range(LIMIT, -1, -1):
result = 0
for indices in itertools.combinations(range(LIMIT), m):
mask = MASK.copy()
for index in indices:
mask[index] = digit
for numbers in itertools.product(range(10), repeat=LIMIT - m):
new_mask = mask.copy()
for number in numbers:
new_mask[new_mask.index('*')] = number
if new_mask[0] == 0 or new_mask.count(digit) != m:
continue
n = 0
for number in new_mask:
n = n * 10 + number
if euler.miller_rabin(n):
result += n
if result:
return result
return 0
F,Phi=Factor(10000001)
#print len(Phi)
def chain(n):
ctr=1
while n > 1:
n = Phi[n]
ctr +=1
return ctr
summ=0
for i in xrange(8900000,10000001):
if miller_rabin(i):
l = chain(i)
if l == 25:
print i,l,F[i]
summ+=i
print "sum is", summ
print time()-st
def main():
return sum(1 for n in range(1, (int((LIMIT / 2)**0.5)))
if euler.miller_rabin(2 * n * n + 2 * n + 1))
def pseudo_fortunate_number(n):
return next(
2 + i
for i in itertools.count()
if euler.miller_rabin(n + 2 + i)
)
#print x
if x<101:
l.append(x)
l = sorted(l)
print l
for k in xrange(2,6):
c = combinations(l,k)
for f in c:
if gcd2(f):
p = sum(f)
if p>100:continue
print k,f,p
if miller_rabin(p):
#print "prime ",f[0],f[1], p
if p not in d :
d[p]=1
else:
d[p]=d[p]+1
summ = 0
for p in d:
if d[p]==1:summ += p;print p
print summ
print
print d
#print sorted(d)
f[i].append(p)
if f[p] == []: f[p] = [p]
return f
from time import time
st = time()
N = 1 * 10**7
print "factoring integers up to ", N
#F=FactorSieve(N)
print "factor sieve complete and hogging memory..."
ctr = 0
plist = []
for j in xrange(100000, 200000):
i = j
if miller_rabin(i): continue
f = RetFact(i)
if len(f) == 1: continue
A = isAchilles(f)
if not A: continue
phiA = Phi(i, f)
#fphi=RetFact(phiA)
fphi = RetFact(phiA)
B = isAchilles(fphi)
if not B: continue
#if A and B and (set(f)==set([2,3]) or set(f)==set([3,2])): print i, f,":",phiA, fphi
#if A and B and (set(f)==set([2,3])): print i, f,",",f.count(2),f.count(3),":",phiA, fphi
#if A and B: print i, len(set(f)),":", f ,":",fphi #,",",f.count(2),f.count(3),":",phiA, fphi
if A and B: print f[-1], "<", i, ":", f, ": ", RetFact(f[-1] - 1)
ctr += 1
plist.append(f[-1])
#print F
print time()-st
exit()
summ=0
x=20000000
y=5000000
z=5000000
for x in xrange(20000000,15000000,-1):
if x%z==0:
v=x/z
z-=1
summ += sum(RetFact(v))
continue
elif miller_rabin(x):
summ += x
continue
else:
summ += sum(RetFact(x))
for k in xrange(0,9):
if j==0 and k>0:break
for m in xrange(0,5): #7
if (k==0 or j==0) and m>0:break
for n in xrange(0,7): #11
if (k==0 or j==0 or m==0) and n>0:break
for o in xrange(0,6): #13
if (k==0 or j==0 or m==0 or n==0) and o>0:break
for p in xrange(0,4): #17
if (k==0 or j==0 or m==0 or n==0 or o==0) and p>0:break
for q in xrange(0,3): #19
if (k==0 or j==0 or m==0 or n==0 or o==0 or p==0) and q>0:break
for r in xrange(0,1): #23
if (k==0 or j==0 or m==0 or n==0 or o==0 or p==0 or q==0) and r>0:break
N = 2**i * 3**j * 5**k * 7**m * 11**n * 13**o * 17**p * 19** q * 23**r;N1=N+1
if N >= 1000000000:continue
while True:
N1 +=2
if miller_rabin(N1):
M = N1-N
#if M not in d:print i,j,k,m,n,o,p,q,r,":",N,M
d[M]=1
#print i,j,k,m,n,o,p,q,r,":",N,M
break
#print i,j,k,m,n,o,p,q,r,":",N
summ=0
for p in d:
#print p
summ+=p
print "sum of distinct pseudo-fortunate numbers is", summ
print "time elapsed is",time()-st
f[i] = [p]
break
if p not in f: f[p] = [p]
return f
st = time()
F = FactorSieve(50000000)
#print F
print time() - st
exit()
summ = 0
x = 20000000
y = 5000000
z = 5000000
for x in xrange(20000000, 15000000, -1):
if x % z == 0:
v = x / z
z -= 1
summ += sum(RetFact(v))
continue
elif miller_rabin(x):
summ += x
continue
else:
summ += sum(RetFact(x))
def check(a, b):
return euler.miller_rabin(int(a + b)) and euler.miller_rabin(int(b + a))
k *= p
f[i].append(p)
if f[p]==[]:f[p]=[p]
return f
from time import time
st = time()
N=1*10**7
print "factoring integers up to ", N
#F=FactorSieve(N)
print "factor sieve complete and hogging memory..."
ctr = 0
plist=[]
for j in xrange(100000,200000):
i = j
if miller_rabin(i):continue
f = RetFact(i)
if len(f)==1:continue
A = isAchilles(f)
if not A:continue
phiA = Phi(i,f)
#fphi=RetFact(phiA)
fphi=RetFact(phiA)
B = isAchilles(fphi)
if not B:continue
#if A and B and (set(f)==set([2,3]) or set(f)==set([3,2])): print i, f,":",phiA, fphi
#if A and B and (set(f)==set([2,3])): print i, f,",",f.count(2),f.count(3),":",phiA, fphi
#if A and B: print i, len(set(f)),":", f ,":",fphi #,",",f.count(2),f.count(3),":",phiA, fphi
if A and B: print f[-1],"<",i,":", f ,": ",RetFact(f[-1]-1)
ctr+=1
plist.append(f[-1])
def main():
return sum(
euler.miller_rabin(3 * i * i + 3 * i + 1)
for i in range(1, int((LIMIT // 3) ** 0.5))
)
if x < 101:
l.append(x)
l = sorted(l)
print l
for k in xrange(2, 6):
c = combinations(l, k)
for f in c:
if gcd2(f):
p = sum(f)
if p > 100: continue
print k, f, p
if miller_rabin(p):
#print "prime ",f[0],f[1], p
if p not in d:
d[p] = 1
else:
d[p] = d[p] + 1
summ = 0
for p in d:
if d[p] == 1:
summ += p
print p
print summ
print
print d