def euler_14():
"""Find the longest sequence using a starting number under one million."""
lengths = {}
def sequence(n):
m = n
length = 1
while m != 1:
if m in lengths:
length += lengths[m]
break
if m % 2 == 0:
m = m / 2
else:
m = 3 * m + 1
length += 1
lengths[n] = length
return length
max_length, max_n = 0, 1
for n in range(1, 1000000):
length = sequence(n)
if length > max_length:
max_length = length
max_n = n
return max_n
def euler_27():
"Find a quadratic formula that produces the maximum number of primes for consecutive values of n."
from itertools import chain, imap, product, takewhile, tee
from eutil import ilen, primes, is_prime, sequence
a_s = xrange(-999, 1000)
ps = tee(takewhile(lambda p: p < 1000, primes()))
b_s = chain(ps[0], imap(lambda x: -x, ps[1]))
seqs = [(ilen(takewhile(is_prime, sequence(lambda n: n ** 2 + a * n + b))), a, b) for a, b in product(a_s, b_s)]
seq = max(seqs, key=lambda t: t[0])
return seq[1] * seq[2]
def euler_65():
"""Find the sum of digits in the numerator of the 100th convergent of the continued fraction for e."""
from eutil import nth, digits
def sequence():
num = []
t = 0
while True:
if t == 0:
num.append(1)
elif t == 1:
num.append(2)
else:
coef = t / 3 * 2 if t % 3 == 0 else 1
num.append(coef * num[t - 1] + num[t - 2])
yield num[-1]
t += 1
return sum(digits(nth(sequence(), 100)))