Created
July 24, 2018 12:00
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A set of very optimised math auxiliar functions that I used in Project Euler.
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from math import sqrt, ceil | |
from itertools import zip_longest, compress, chain, product, combinations | |
from functools import reduce | |
def prime_sieve(n): | |
""" | |
Returns: list of primes, 2 <= p < n | |
Performance: 0.5s for 10**7, 5s for 10**8, 69s for 10**9 | |
""" | |
sieve = [True] * int(n / 2) | |
for i in range(3, ceil(sqrt(n)), 2): | |
if sieve[int(i / 2)]: | |
sieve[int(i * i / 2)::i] = [False] * int((n - i * i - 1) / (2 * i) + 1) | |
return [2] + [2 * i + 1 for i in range(1, int(n / 2)) if sieve[i]] | |
def is_prime(n): | |
""" | |
Returns: whether a number is prime or not | |
Performance: 4.5s for 10**16, 15s for 10**17, 46.2s for 10**18 | |
""" | |
res = True | |
if n == 2 or n == 3: | |
res = True | |
elif n < 2 or n % 2 == 0: | |
res = False | |
elif n < 9: | |
res = True | |
elif n % 3 == 0: | |
res = False | |
r = int(sqrt(n)) | |
f = 5 | |
while f <= r: | |
if n % f == 0 or n % (f + 2) == 0: | |
res = False | |
else: | |
f += 6 | |
return res | |
def fibonacci(n): | |
""" | |
Returns: the nth fibonacci number | |
Performance: 0.1s for 10**5, 0.8s for 10**6, 65.4s for 10**7 | |
""" | |
return _fib(n)[0] | |
def _fib(n): | |
if n == 0: | |
return (0, 1) | |
else: | |
a, b = _fib(n // 2) | |
c = a * (2 * b - a) | |
d = b * b + a * a | |
if n % 2 == 0: | |
return (c, d) | |
else: | |
return (d, c + d) | |
def lcm(a, b): | |
return a * b / gcd(a, b) | |
def gcd(a, b): | |
if a < 0: a = -a | |
if b < 0: b = -b | |
if a == 0: return b | |
while (b): a, b = b, a % b | |
return a | |
def factor(n): | |
""" | |
Returns: the factors of n and its exponents | |
Performance: 0.1 for 10**10**3, 0.3s for 10**10**4, 29.1s for 10**10**5 | |
""" | |
f, factors, prime_gaps = 1, [], [2, 4, 2, 4, 6, 2, 6, 4] | |
if n < 1: | |
return [] | |
while True: | |
for gap in ([1, 1, 2, 2, 4] if f < 11 else prime_gaps): | |
f += gap | |
if f * f > n: # If f > sqrt(n) | |
if n == 1: | |
return factors | |
else: | |
return factors + [(n, 1)] | |
if not n % f: | |
e = 1 | |
n //= f | |
while not n % f: | |
n //= f | |
e += 1 | |
factors.append((f, e)) | |
def pal_list(k): | |
""" | |
Returns: list of al palindromic numbers with k digits | |
Performance: 0.1s for 8, 0.5s for 10, 59.4s for 14 | |
""" | |
if k == 1: | |
return [1, 2, 3, 4, 5, 6, 7, 8, 9] | |
return [sum([n * (10**i) for i, n in enumerate(([x] + list(ys) + [z] + list(ys)[::-1] + [x]) if k % 2 else ([x] + list(ys) + list(ys)[::-1] + [x]))]) | |
for x in range(1, 10) for ys in product(range(10), repeat=k // 2 - 1) for z in (range(10) if k % 2 else (None,))] | |
def phi(n): | |
res = n | |
factors = [f[0] for f in factor(n)] | |
multiplier = 1 | |
for fs in range(1, len(factors) + 1): | |
multiplier *= -1 | |
for primes in combinations(factors, fs): | |
val = reduce(lambda x, y: x * y, primes) | |
if val <= n and n % val == 0: | |
res += (n // val) * multiplier | |
return res |
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