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from fractions import Fraction
from math import gcd


# compute the sequence OEIS A092317
#    https://oeis.org/A092317
def _slowway(n):
    b = Fraction(0)
    i = 3
    while b < n:

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# The MIT License (MIT)
#
# Copyright (c) 2022 Neil Webber
#
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:

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# MIT License
#
# Copyright (c) 2022,2023 Neil Webber
#
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:

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# This implements the "Knights and Pages" problem
# from The Chicken from Minsk, Chapter 1
#
# PROBLEM:
#   Many years ago three knights waited to cross the river Neva.
#   Each knight had his own page, so there were six people. The
#   boat they had could only carry two. However, the knights were
#   ferocious killers and the pages were terrified. In fact it was
#   certain that any one of the pages would die of heart failure
#   if he were not protected at every instant from the other knights

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class SetattrThenRestore:
    """Save the old value of an attribute, set it, then restore it.

    with SetattrThenRestore(obj, attrname, val):
        ...bunch of code here...

    is SOMEWHAT equivalent to:

        oldattrval = getattr(obj, attrname)
        setattr(obj, attrname, val)

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# miscellaneous descriptor-protocol classes, implemented mostly as an exercise
#

# The MIT License
#
# Copyright (c) 2022 Neil Webber
#
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights

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from itertools import combinations, chain
from math import prod


def _powA258484(i, n):
    """Return i**n for two integers i and n."""

    # NOTE: Putting in these special cases sped up the search
    #       process enough to be worth it (measured via timeit)
    if n == 0:

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from functools import lru_cache

@lru_cache
def _powA046253(i, n):
    """Return i**n for two integers i and n. Defines 0**0 to be ZERO."""
    r = i
    while n > 1:
        r *= i
        n -= 1
    return r

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# toy example re-implementing C3 method resolution order
# (MRO, as used for python multi-inheritance)
#
# Algorithm snippets in comments come from:
#      https://www.python.org/download/releases/2.3/mro/
# and are notated as [2.3MRO]
#

# A Hier represents a class hierarchy, with a name, and a possibly-empty
# list of bases (each base itself also a Hier).

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#!/usr/bin/env python3

def collatz_sequence(n):
    """Generate the collatz sequence for n (see OEIS A006577)."""

    c = int(n)
    if c < 1 or c != n:
        raise ValueError(f"Argument n ({n}) must be a positive integer")

    while c != 1: