def test(sols=10**3, np=2, start=2, test=100):
t = time()
n = 2
z = 2
primes = 2
while z < np:
primes *= z
z = nextprime(z)
while True:
for i in range(start, primes * np + 1, primes):
#count = numfacts(i**2)//2+1
count = numdivs(i**2) // 2 + 1
if count > test:
print('start: ',i,'time:',time()-t,'counts:',count,\
'test:',test,'next:',np)
test += (count // 100 * 100 - test + 100)
if count > sols:
print(time() - t)
print('n: ', i, 'time:',
time() - t, 'counts:', count)
test += (count // 100 * 100 - test + 100)
return i
primes *= np
np = nextprime(np)
start = primes
def e146(top=10 ** 6):
ret = 0
x1 = lambda n: n ** 2 + 1
x2 = lambda n: n ** 2 + 3
x3 = lambda n: n ** 2 + 7
x4 = lambda n: n ** 2 + 9
x5 = lambda n: n ** 2 + 13
x6 = lambda n: n ** 2 + 27
test = [x1, x2, x3, x4, x5, x6]
mini = 1000
for i in range(1, top + 1):
if i > mini:
print(i)
mini += 1000
z = 0
yesno = i
for j in test:
y = j(i)
if z:
if y != z:
yesno = 0
break
if not isprime2(y):
continue
z = nextprime(y)
ret += yesno
return ret
def e146(top=10**6):
ret = 0
x1 = lambda n: n**2 + 1
x2 = lambda n: n**2 + 3
x3 = lambda n: n**2 + 7
x4 = lambda n: n**2 + 9
x5 = lambda n: n**2 + 13
x6 = lambda n: n**2 + 27
test = [x1, x2, x3, x4, x5, x6]
mini = 1000
for i in range(1, top + 1):
if i > mini:
print(i)
mini += 1000
z = 0
yesno = i
for j in test:
y = j(i)
if z:
if y != z:
yesno = 0
break
if not isprime2(y): continue
z = nextprime(y)
ret += yesno
return ret
def test(sols = 10**3,np=2,start=2,test = 100):
t = time()
n = 2
z = 2
primes = 2
while z<np:primes*=z;z = nextprime(z)
while True:
for i in range(start,primes*np+1,primes):
#count = numfacts(i**2)//2+1
count = numdivs(i**2)//2+1
if count>test:
print('start: ',i,'time:',time()-t,'counts:',count,\
'test:',test,'next:',np)
test+=(count//100*100-test+100)
if count>sols:
print(time()-t)
print('n: ',i,'time:',time()-t,'counts:',count);test+=(count//100*100-test+100)
return i
primes *=np
np = nextprime(np)
start = primes
def factors(n):
global primes
p = primes
if n ==1:return [1]
if n in p:return [n]
else:
for i in range(len(p)):
if n%p[i]==0:
result =[p[i]]+factors(n//p[i])#,primes)
return result
if isprime2(n):return [n]
else:
primes+=[nextprime(primes[-1])]
return factors(n)#,primes)
def factors(n):
global primes
p = primes
if n == 1:
return [1]
if n in p:
return [n]
else:
for i in range(len(p)):
if n % p[i] == 0:
result = [p[i]] + factors(n // p[i]) # ,primes)
return result
if isprime2(n):
return [n]
else:
primes += [nextprime(primes[-1])]
return factors(n) # ,primes)
def e108(sols = 10**3,p = True):
t = time()
n = 2
np = 1
primes = 2
test = 100
while True:
np = nextprime(np)
for i in range(primes,primes*np+1,primes):
if p:count = numfacts(i**2)//2+1
else:count = numdivs(i**2)//2+1
if count>test:
print('n: ',i,'time:',time()-t,'counts:',count,'test: ',test)
test+=(count//100*100-test+100)
if count>sols:
print(time()-t)
print('n: ',i,'time:',time()-t,'counts:',count);test+=(count//100*100-test+100)
return i
primes *=np
def solve():
p = 1
while True:
p = nextprime(p)
digs = digits(p)
numdigs = len(digs)
uniquedigs = list(set(digs))
uniquedigs.sort()
indexes = [None] * 10
for i in range(numdigs):
d = digs[i]
if indexes[d] is None:
indexes[d] = []
indexes[d].append(i)
for d in uniquedigs:
# The first member of a family have recurring digits,
# each of which is one of 0, 1, and 2.
if d > 2:
continue
inds = indexes[d]
# If the number of recurring digits is not a multiple of 3 or
# one of the digits is the least significant, then the family
# can't have 8 members.
if len(inds) % 3 == 0 and inds[-1] != numdigs - 1:
start = 1 if inds[0] == 0 else 0
numgenerated = 10 - start
numprimes = 0
numcomposites = 0
# Generate family candidates
for e in range(start, 10):
for i in inds:
digs[i] = e
number = digits_to_number(digs)
if isprime(number):
numprimes += 1
else:
numcomposites += 1
if numcomposites > numgenerated - 8:
break
if numprimes == 8:
return p
def solve():
p = 1
while True:
p = nextprime(p)
digs = digits(p)
numdigs = len(digs)
uniquedigs = list(set(digs))
uniquedigs.sort()
indexes = [None] * 10
for i in range(numdigs):
d = digs[i]
if indexes[d] is None:
indexes[d] = []
indexes[d].append(i)
for d in uniquedigs:
# The first member of a family have recurring digits,
# each of which is one of 0, 1, and 2.
if d > 2:
continue
inds = indexes[d]
# If the number of recurring digits is not a multiple of 3 or
# one of the digits is the least significant, then the family
# can't have 8 members.
if len(inds) % 3 == 0 and inds[-1] != numdigs - 1:
start = 1 if inds[0] == 0 else 0
numgenerated = 10 - start
numprimes = 0
numcomposites = 0
# Generate family candidates
for e in range(start, 10):
for i in inds:
digs[i] = e
number = digits_to_number(digs)
if isprime(number):
numprimes += 1
else:
numcomposites += 1
if numcomposites > numgenerated - 8:
break
if numprimes == 8:
return p
def e108(sols=10**3, p=True):
t = time()
n = 2
np = 1
primes = 2
test = 100
while True:
np = nextprime(np)
for i in range(primes, primes * np + 1, primes):
if p: count = numfacts(i**2) // 2 + 1
else: count = numdivs(i**2) // 2 + 1
if count > test:
print('n: ', i, 'time:',
time() - t, 'counts:', count, 'test: ', test)
test += (count // 100 * 100 - test + 100)
if count > sols:
print(time() - t)
print('n: ', i, 'time:',
time() - t, 'counts:', count)
test += (count // 100 * 100 - test + 100)
return i
primes *= np
