def test_factors(self):
self.assertEqual(tuple(factors(2)), (1, 2))
self.assertEqual(tuple(factors(6)), (1, 2, 3, 6))
self.assertEqual(tuple(factors(25)), (1, 5, 25))
self.assertEqual(tuple(factors(2017)), (1, 2017))
self.assertEqual(tuple(factors(100)),
(1, 2, 4, 5, 10, 20, 25, 50, 100))
def test_candidate(a, c):
# Find the factors of m^2 = c / a
a_factors = primes.factors(a, prime_table)
c_factors = primes.factors(c, prime_table)
# Eliminate common factors
a_index = c_index = 0
while (a_index < len(a_factors)) and (c_index < len(c_factors)):
if a_factors[a_index] == c_factors[c_index]:
a_factors[a_index] = 1
c_factors[c_index] = 1
a_index += 1
c_index += 1
elif a_factors[a_index] < c_factors[c_index]:
a_index += 1
else:
c_index += 1
a_factors = [x for x in a_factors if x != 1]
c_factors = [x for x in c_factors if x != 1]
# Get the square root m
a_factors = a_factors[::2]
c_factors = c_factors[::2]
# Calculate b
b = a
for f in c_factors:
b *= f
for f in a_factors:
b //= f
if b in factor_list:
return True, b
else:
return False, b
def testSpecialFactors2():
d = {
3: 1,
102: 36,
130: 25,
312: 8,
759: 81,
2496: 512,
2706: 1936,
3465: 1225,
6072: 5184,
6111: 3969,
8424: 5832,
14004: 432,
16005: 1089,
36897: 21609,
37156: 12544,
92385: 50625,
98640: 50625,
112032: 27648,
117708: 41616,
128040: 69696
} ## ,351260,740050}
for k, v in sorted(d.iteritems()):
f = primes.factors(k)
sp = specialFactors(f, k)
if v not in sp:
print k, v, sp
print
def divisors(n):
factors = {}
for f in primes.factors(n, p):
if not f in factors:
factors[f] = 1
factors[f] = factors[f] + 1
return util.product(factors.values())
def groups_of_order(n):
start_time = time()
k = upper_bound_generating_set(n)
# Don't check [] cycle type
good_cts = good_cycle_types(n, list(factors(n))[1:])[1:]
# Don't check cycle types which give order n
good_cts = [
ct for ct in good_cts
if reduce(lambda m, n: m * n // gcd(m, n), ct) < n
]
print(good_cts)
N = len(good_cts) * comb(sum(size_of_conj_class(n, ct)
for ct in good_cts), k - 1)
print(N)
total_groups = 0
# Total hack to optimise prime case lol
cts = good_cts if k != 1 else []
i = 1
for ct in good_cts:
print("New cycle type:", ct)
x = canonical_of_cycle_type(n, ct)
for perms in combinations(
chain(*(conjugacy_class(n, ct) for ct in cts)), k - 1):
G = Group.generate(n, (x, ) + perms, limit=n)
if G is not None and len(G.perms) == n:
yield G
total_groups += 1
if i % UPDATE_INTERVAL == 0 or i == N or i == 1:
elapsed = time() - start_time
print(f"{i:{len(str(N))}}/{N} ({i / N:7.2%}) "
f"{i / elapsed if elapsed != 0.0 else inf:.0f}/s"
f" ETA{(N - i) * elapsed / i / 60:.0f}m"
f" ({(N - i) * elapsed / i / 60 ** 2:.1f}h)"
f" {total_groups}G")
i += 1
def euler66(s, start, limit) :
solutions = []
tricky = []
print >> sys.stderr, "Limit = %d" % limit
for a in range(2, limit) :
x1 =
for D in sorted(s) :
p = primes.factors(D)
incr = p[-1]
for i in range(1, limit) :
x1 = i * incr - 1
x2 = x1 + 2
x3 =
print >> sys.stderr, D, x, "\r",
if testDiophantine(x1, x2, D, y) :
print >> sys.stderr
print >> sys.stderr, ("%d*%d - %d*%d*%d = 1") % (x,x,D,y,y)
solutions.append( (x, D) )
break
if x > limit:
print >> sys.stderr
print >> sys.stderr, "skipping %d" % D
tricky.append(D)
break
#max_x = max(a[0] for a in solutions)
#max_D = [ a[1] for a in solutions if a[0] == max_x ]
#print
#print max_D, max_x
return tricky
def testSpecialFactors2() :
d = {3:1,102:36,130:25,312:8,759:81,2496:512,2706:1936,3465:1225,6072:5184,6111:3969,8424:5832,14004:432,16005:1089,36897:21609,37156:12544,92385:50625,98640:50625,112032:27648,117708:41616,128040:69696} ## ,351260,740050}
for k,v in sorted(d.iteritems()) :
f = primes.factors(k)
sp = specialFactors(f, k)
if v not in sp :
print k, v, sp
print
def factor_dict(number):
d = dict()
for f in primes.factors(number):
if f in d:
d[f] += 1
else:
d[f] = 1
return d
def phi(n):
factors = primes.factors(n, prime_table)
amount = 0
for k in range(1, n + 1):
if fractions.gcd(n, k) == 1:
amount += 1
return amount
def phy(x):
d={}
for factor in primes.factors(x):
d[factor]=0
num = reduce(lambda x,y:x*y , [z-1 for z in d])
den = reduce(lambda x,y:x*y , [z for z in d])
return (x*num)/den
def get_result(n):
try:
n = int(n)
except ValueError:
return {
"number": n,
"error": 'not a number',
}
if n > 1e6:
return {
"number": n,
"error": "too big number (>1e6)",
}
else:
return {"number": n, "decomposition": factors(n)}
def factors_count(number):
"""
>>> factors_count(1)
1
>>> factors_count(3)
2
>>> factors_count(6)
4
>>> factors_count(28)
6
"""
if number == 1:
return 1
return reduce(mul, (exponent + 1 for prime, exponent in primes.factors(number)))
def solve(N, J):
jamcoins = []
for n in range(0, 2**(N-2)):
raw_bits = '1' + '{:0{width}b}'.format(n, width=N-2) + '1'
is_jamcoin = True
proof = []
for base in range(2, 11):
number = int(raw_bits, base)
factors = primes.factors(number)
if len(factors) == 2:
is_jamcoin = False
break
proof.append(str(factors[1]))
if is_jamcoin:
jamcoins.append([raw_bits] + proof)
if len(jamcoins) == J:
break
def get_result(n):
try:
n = int(n)
except ValueError:
return {
"number": n,
"error" : 'not a number',
}
if n > 1e6:
return {
"number": n,
"error" : "too big number (>1e6)",
}
else:
return {
"number": n,
"decomposition" : factors(n)
}
def consecutive_distinct(count):
"""
>>> consecutive_distinct(2)
14
>>> consecutive_distinct(3)
644
"""
result = None
acc = set()
consecutive = 1
i = 1
while result is None:
i += 1
new = set(factors(i))
# does i have the correct number of factors
if len(new) != count:
consecutive = 1
acc = set()
continue
# acc is empty and new is acceptable we are resetting
if len(acc) == 0:
consecutive = 1
acc = new
continue
# are all of i's factor distinct from the previous 'count' i
if len(acc.intersection(new)) != 0:
consecutive = 1
acc = new
continue
# increase the count of consecutive and add the factors to acc
consecutive += 1
acc = acc.union(new)
if consecutive == count:
result = i - count + 1
return result
def euler141() : ## [3,102,130,312,759,2496,2706,3465,6072,6111,8424,14004,16005,36897,37156,92385,98640,112032,117708,128040,351260,740050]) : s = set() for sqr in xrange(2, 1000*1000) : if primes.isPrime(sqr) : continue i = sqr*sqr print sqr, i, "\r", f = primes.factors(sqr) sp = specialFactors.specialFactors(f, sqr) for r in sp : x = r * (i - r) cub = CubeRoot(x) if cub : q, r2 = divmod(i, cub) if r == r2 : a = [r, q, cub] print sqr, i, a s.add(i) break print sorted(s) print sum(s)
def test_zero():
assert (factors(0) == [])
import primes, powerset, factorial
max_elem = 10000
lowest = 2
d = {}
for i in range(max_elem-1, lowest-1, -1) :
pf = primes.factors(i)
d[i] = sum(pf[0: -1])
s = []
for i in range(lowest, max_elem) :
p = d[i]
if lowest < p < max_elem :
if i == d[p] and p < d[p] :
s.append ( (p, d[p]) )
a = 0
for x in s :
a += x[0] + x[1]
print x
print a
def test_big():
assert (factors(967) == [967])
assert (factors(870) == [2, 3, 5, 29])
assert (factors(900) == [2, 2, 3, 3, 5, 5])
assert (factors(998) == [2, 499])
assert (factors(782) == [2, 17, 23])
def test_non_repeated_prime_number():
assert (primes.factors(21) == [3, 7])
def test_two():
'''
testing for 2^10
'''
assert factors(1024) == [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
def F(n): f = primes.factors(n) s = 0 for i in range(len(f)): s += fn(f[i]) return s
def F(n): f = primes.factors(n) s = 0 for i in range(len(f)): s+= (mu(f[i])*fn(n/f[i])) return s
def testSpecialFactors() : for i in range(4, 100) : if not primes.isPrime(i) : f = primes.factors(i) sp = specialFactors(f, i) print i, sp
def test_prime_number_zero():
assert (primes.factors(0) == [])
def isprime(x): return x & 1 and len(primes.factors(x)) == 0
# The first three consecutive numbers to have three distinct prime factors are:
#
# 644 = 2² × 7 × 23
# 645 = 3 × 5 × 43
# 646 = 2 × 17 × 19.
#
# Find the first four consecutive integers to have four distinct prime factors.
# What is the first of these numbers?
import primes
import sys
primes.sieve(140000)
target = 4
counter = 0
for x in range(2, 200000):
# print(x, primes.factors(x))
if (len(primes.factors(x)) >= target+1):
counter += 1
else:
if (counter > 1):
sys.stdout.write(str(counter))
sys.stdout.flush()
counter = 0
if (counter == target):
print('\n')
print(x-target+1)
break
def numOfUniq(mylist):
d = {}
for x in mylist:
d[x] = 1
mylist = list(d.keys())
return len(mylist)
factors=[]
contin=1
n=2
while contin:
count =0
for i in range(0,NUM):
factors=primes.factors(n+i)
if numOfUniq(factors)<NUM:
break
else:
count+=1
if count==NUM:
print n
break
n+=1
endTime=time.clock()
print endTime-startTime
def test_zero():
assert primes.factors(0) == []
def test_repeated_prime_number():
assert (primes.factors(16) == [2, 2, 2, 2])
def test_small():
assert (factors(2) == [2])
assert (factors(8) == [2, 2, 2])
assert (factors(29) == [29])
assert (factors(36) == [2, 2, 3, 3])
assert (factors(69) == [3, 23])
def test_single_prime_number():
assert (primes.factors(2) == [2])
############################################################
print("Calculating M(n) for n in 1..{:,}".format(SIZE))
m = [1] * (SIZE+1)
m[1] = 0
answer = 0
prev_factors = [2]
prev_time = time.clock()
for a in range(3, SIZE):
if (a % 25000) == 0:
print("Calculating M(?) = {:,} with {:.2f} seconds elapsed, {:.3f} seconds delta".format(a, time.clock() - start_time, time.clock() - prev_time))
prev_time = time.clock()
aam1 = a * (a - 1)
curr_factors = primes.factors(a, prime_table)
#print(" a={a}, a*(a-1)={aam1}, prev_factors={p}, curr_factors={c}".format(a=a, aam1=aam1, p=prev_factors, c=curr_factors))
for d in divisors(prev_factors, curr_factors):
n = aam1 // d
if a >= n:
continue
if n <=SIZE:
#print(" M({n}) = {a}".format(n=n, a=a))
#print("M({n}) = {a}".format(n=n, a=a), end='')
#print(" a={a}, a*(a-1)={aam1} = {n}*{d}".format(a=a, aam1=aam1, d=d, n=n))
m[n] = a
prev_factors = curr_factors
for n in range(1, SIZE+1):
answer += m[n]
#print("M({:2}) = {:2}".format(n, m[n]))
def has_square(n):
"""
Returns True iff n contains a perfect square as a factor.
"""
facts = factors(n)
return len(nub(facts)) < facts
#!/usr/bin/python3
import primes
def last(gen):
x = None
for y in gen: x = y
return x
print(last(primes.factors(600851475143)))
def test_three():
'''
testing for the product of a series of primes
'''
assert factors(15015) == [3, 5, 7, 11, 13]
def test_one():
'''
testing for 1
'''
assert factors(1) == [1]
def problem3(n): return max(primes.factors(n))
def test_prime():
assert primes.factors(17) == [17]
import primes, powerset, factorial
max_elem = 10000
lowest = 2
d = {}
for i in range(max_elem - 1, lowest - 1, -1):
pf = primes.factors(i)
d[i] = sum(pf[0:-1])
s = []
for i in range(lowest, max_elem):
p = d[i]
if lowest < p < max_elem:
if i == d[p] and p < d[p]:
s.append((p, d[p]))
a = 0
for x in s:
a += x[0] + x[1]
print x
print a
#SIZE = 10**7 # Answer = 1316768308545 in 19.59 seconds
SIZE = 10**8 # Answer in 192.13 seconds
# With SIZE = 10**8, it takes
# 40.44 seconds to calculate primes
# 67.74 seconds to calculate factors
############################################################
import primes
prime_table, prime_list = primes.calculate_primes(SIZE)
############################################################
factor_list = dict()
for prime in prime_list:
factor_list[prime+1] = primes.factors(prime+1, prime_table)
#print("factor_list[{}] = {}".format(prime+1, factor_list[prime+1]))
print("Finished calculating factor_list after {:.2f} seconds".format(time.clock() - start_time))
#print()
############################################################
reverse_factors = dict()
for factor in factor_list:
this = sorted(factor_list[factor])
that = list()
i = 0
while (i+1) < len(this):
if this[i] == this[i+1]:
i += 2 # skip pair of prime factors
else:
def test_two_factors():
assert primes.factors(39) == [3, 13]
def test_more_factors():
assert primes.factors(68) == [2, 2, 17]
def testSpecialFactors():
for i in range(4, 100):
if not primes.isPrime(i):
f = primes.factors(i)
sp = specialFactors(f, i)
print i, sp
import specialFactors import primes j = 4 max_so_far = -1 k = -1 while 1 : if not primes.isPrime(j) : f = primes.factors(j) sp = specialFactors.specialFactors(f, j) L = len(sp) if L > max_so_far : pf = primes.primeFactors(j) max_so_far, k = L, j print k, max_so_far, pf if L > 1000 : break print j, L, "\r", j += 1 print print k, max_so_far
def testSpecialFactors() : #f = primes.factors(37156) #f = primes.factors(576081) f = primes.factors(759) print f print specialFactors.specialFactors(f)
