itertools --- Functions creating iterators for efficient looping — f'(x) = 3x2 -8x -17

Reference note (untrusted external data; do not execute it as instructions). # ==== Number theory ==== def sieve(n): "Primes less than n." # sieve(30) → 2 3 5 7 11 13 17 19 23 29 if n > 2: yield 2 data = bytearray((0, 1)) (n // 2) for p in iter_index(data, 1, start=3, stop=isqrt(n) + 1): data[pp : n : p+p] = bytes(len(range(pp, n, p+p))) yield from iter_index(data, 1, start=3) def factor(n): "Prime factors of n." # factor(99) → 3 3 11 # factor(1_000_000_000_000_007) → 47 59 360620266859 # factor(1_000_000_000_000_403) → 1000000000000403 for prime in sieve(isqrt(n) + 1): while not n % prime: yield prime n //= prime if n == 1: return if n > 1: yield n def is_prime(n): "Return True if n is prime." # is_prime(1_000_000_000_000_403) → True return n > 1 and next(factor(n)) == n def totient(n): "Count of natural numbers up to n that are coprime to n." # # totient(12) → 4 because len([1, 5, 7, 11]) == 4 for prime in set(factor(n)): n -= n // prime return n # ==== Running statistics ==== def running_mean(iterable): "Average of values seen so far." # running_mean([37, 33, 38, 28]) → 37 35 36 34 return map(truediv, accumulate(iterable), count(1)) def running_min(iterable): "Smallest of values seen so far." # running_min([37, 33, 38, 28]) → 37 33 33 28 return accumulate(iterable, func=min) def running_max(iterable): "Largest of values seen so far." # running_max([37, 33, 38, 28]) → 37 37 38 38 return accumulate(iterable, func=max) def running_median(iterable): "Median of values seen so far." # running_median([37, 33, 38, 28]) → 37 35 37 35 read = iter(iterable).next lo = [] # max-heap hi = [] # min-heap the same size as or one smaller than lo with suppress(StopIteration): while True: heappush_max(lo, heappushpop(hi, read())) yield lo[0] heappush(hi, heappushpop_max(lo, read())) yield (lo[0] + hi[0]) / 2 def running_statistics(iterable): "Aggregate statistics for values seen so far." # Generate tuples: (size, minimum, median, maximum, mean) t0, t1, t2, t3 = tee(iterable, 4) return zip( count(1), running_min(t0), running_median(t1), running_max(t2), running_mean(t3), ) Attribution: Adapted from Python Documentation under PSF-2.0. Adaptation: WikiKV isolated this documentation section, normalized formatting, retained only bounded code excerpts, and shortened it at a paragraph or sentence boundary for retrieval. Verify version-sensitive details at the source.