def get_factors(x):
global cached_factors
if x in cached_factors:
return cached_factors[x]
facs = factors(x)
cached_factors[x] = facs
return facs
def totient(n):
factors = euler.factors(n)
coprime = [1 for i in range(n)]
coprime[0] = 0
for f in factors:
for i in range(f, n, f):
coprime[i] = 0
return sum(coprime)
# ####################################################
# Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n).
# If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and each of a and b are called amicable numbers.
# For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110; therefore d(220) = 284. The proper divisors of 284 are 1, 2, 4, 71 and 142; so d(284) = 220.
# Evaluate the sum of all the amicable numbers under 10000.
####################################################
from euler import factors
divisor_sums = {}
max_value = 10000
for i in range(2, max_value):
divisor_sums[i] = int(sum(factors(i)) - i)
amicable_numbers = set()
for i in range(2, max_value):
j = divisor_sums[i]
# If we've already found this value from its partner
if i in amicable_numbers:
continue
if j not in divisor_sums.keys():
divisor_sums[j] = int(sum(factors(j)) - j)
if divisor_sums[j] == i and i != j:
if j > 0 and j < max_value:
amicable_numbers.add(j)
# Mar 29, 2016
import euler
n, new_num = 1, 1
while len(euler.factors(new_num)) < 500:
n += 1
new_num = int((n * (n + 1)) / 2)
print "Ans:", new_num
####################################################
# The prime factors of 13195 are 5, 7, 13 and 29.
# What is the largest prime factor of the number 600851475143?
####################################################
from euler import is_prime, factors
# Get the factors of x
def get_factors(x):
global cached_factors
if x in cached_factors:
return cached_factors[x]
facs = factors(x)
cached_factors[x] = facs
return facs
# Do the calculation
# To avoid redundant calculations, we cache answers
cached_factors = {}
facs = sorted(factors(600851475143), reverse=True)
for x in facs:
if is_prime(x):
print(x)
break
def f(n):
# return n/float(totient(n))
return n / float(totient2(n))
def main(maxv):
best = 2
f_best = 0
for n in range(390390, maxv):
if euler.isprime(n) == False:
f_n = f(n)
if f_n > f_best:
print 'possible', n, f_n
best = n
f_best = f_n
if n % 1000 == 0:
print '.' #n,f(n),'best',best,f_best
#main(1000000)
for i in range(2, 1000000):
if euler.isprime(i) == False:
if len(euler.factors(i)) > 50:
if f(i) > 5.1:
print i, len(euler.factors(i)), f(i)
#v = totient2(i)
#if i % 1000 == 0:
# print i,v,totient(i),i/float(v),len(euler.factors(i))
#!/usr/bin/python import euler print(euler.factors(600851475143))
primeFactors = euler.primeFactors(230, primes)
assert primeFactors == primeFactors0
primeFactors0 = [2, 2, 3, 3]
primeFactors = euler.primeFactors(36, primes)
assert primeFactors == primeFactors0
primePowerFactorization0 = {2: 2, 3: 2}
primePowerFactorization = euler.primePowerFactorization(primeFactors)
assert primePowerFactorization0 == primePowerFactorization
assert (euler.descendingFactorial(5, 2) == 20)
assert (euler.descendingFactorial(5, 3) == 60)
assert (euler.ascendingFactorial(5, 2) == 30)
assert (euler.ascendingFactorial(5, 3) == 210)
assert (euler.descendingFactorial(5, -2) == 1 / 42)
assert (euler.descendingFactorial(5, -3) == 1 / 336)
assert (euler.ascendingFactorial(5, -2) == 1 / 12)
assert (euler.ascendingFactorial(5, -3) == 1 / 24)
assert (euler.factors(20, []) == [1, 2, 4, 5, 10, 20])
assert (euler.appendDigits(37465, []) == [3, 4, 5, 6, 7])
assert (euler.appendDigitsUnsorted(37465, []) == [3, 7, 4, 6, 5])
primes = euler.readPrimes()
assert (primes[0:20] == [
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71
])
def testFactors(self):
self.assert_(euler.factors(24) == set([2,3,4,6,8,12]))
# April 15, 2016
import euler
amicable_numbers = list()
for n in range(1, 10000):
b = sum(euler.factors(n)[:-1])
if n != b and sum(
euler.factors(b)[:-1]) == n and n not in amicable_numbers:
amicable_numbers.append(n)
if b <= 10000:
amicable_numbers.append(b)
print sorted(amicable_numbers)
print '\n' + str(sum(amicable_numbers))
#!/usr/bin/python
import euler
p = 0
n = 1
while p < 10001:
n += 1
if (n % 2 == 0) or (n % 3 == 0) or (n % 5 == 0):
continue
if len(euler.factors(n)) == 1:
p += 1
print(n)
# Let us list the factors of the first seven triangle numbers:
# 1: 1
# 3: 1,3
# 6: 1,2,3,6
# 10: 1,2,5,10
# 15: 1,3,5,15
# 21: 1,3,7,21
# 28: 1,2,4,7,14,28
# We can see that 28 is the first triangle number to have over five divisors.
# What is the value of the first triangle number to have over five hundred divisors?
####################################################
from euler import factors
def triangle_number(n):
return n * (n + 1) // 2
n = 1
while True:
tri_num = triangle_number(n)
facs = factors(tri_num)
if len(facs) > 500:
break
n += 1
print(tri_num)
#!/usr/bin/python
import euler
total_factors = [0] * 21
for n in range(20, 2, -1):
f = euler.factors(n)
while f:
x = f[0]
c = f.count(x)
total_factors[x] = max(total_factors[x], c)
f = [e for e in f if e != x]
prod = 1
for x in range(2, 21):
prod *= (x**total_factors[x])
print(prod)
def primeFactors(n):
s = set()
for f in euler.factors(n):
if euler.isprime(f):
s.add(f)
return s
#!/usr/bin/python
import euler
tn = 1
c = 1
ld = 1
while ld < 500:
c += 1
tn += c
f = euler.factors(tn)
if (len(f) > 8):
d = euler.divisors(f)
ld = len(d)
print(c, tn, ld)
import euler
d = [sum(list(euler.factors(i))) + 1 for i in range(10001)]
s = set()
for a in range(10001):
try:
b = d[a]
if d[b] == a and a != b:
s.add((min(a, b), max(a, b)))
except:
pass
lst = []
for a, b in s:
lst.append(a)
lst.append(b)
print lst
print sum(lst)
def divisors(n):
lst = [1] + list(euler.factors(n))
lst.sort()
return lst
import euler
for i in range(10,100):
for j in range(10,100):
if i != j and ((i%10!=0) and (j%10!=0)):
ifactors = euler.factors(i)
jfactors = euler.factors(j)
if ifactors.intersection(jfactors):
print i,'/',j,' ',
def factors(n):
s = euler.factors(n)
s.add(1)
s.add(n)
return list(s)
def is_abundant(n):
facs = factors(n)
div_sum = sum(facs) - n
return div_sum > n
def wrapper(*args):
s = func(*args)
if is_prime(args[0]):
return set([1])
return s.difference(args)
return wrapper
def sum_it(val):
if len(val) < 2:
return 1
return reduce(lambda x, y: x + y, val)
factors = remove_last(factors)
# print sorted(factors(220))
# print reduce(lambda x, y: x+y, sorted(factors(220)))
# print reduce(lambda x, y: x+y, sorted(factors(284)))
sum = 0
for num in range(2, 10000):
a = sum_it(factors(num))
# print a
# print factors(a)
b = sum_it(factors(a))
# print b
if num == b and num != a:
print "here", num
sum += num
print "the answer is: ", sum
# April 17, 2016
import euler
limit = 28123
abundant_numbers = list()
for n in xrange(12, limit + 1):
if sum(euler.factors(n)[:-1]) > n:
abundant_numbers.append(n)
all_sums = list()
for j in range(len(abundant_numbers) - 1):
for k in range(j, len(abundant_numbers) - 1):
if abundant_numbers[j] + abundant_numbers[k] > limit:
break
all_sums.append(abundant_numbers[j] + abundant_numbers[k])
all_sums = sorted(list(set(all_sums)))
ans, k = 0, 0
for n in range(1, limit + 1):
if n not in all_sums[k:]:
ans += n
if n > all_sums[k] and k < len(all_sums) - 1:
k += 1
print ans