eulerbooks/notebooks/problem0014.ipynb

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    "# [Longest Collatz Sequence](https://projecteuler.net/problem=14)\n",
    "\n",
    "The [Collatz conjecture](https://en.wikipedia.org/wiki/Collatz_conjecture) is a famous unsolved problem, and a classic example of a seemingly simple question that has proven very difficult, if not impossible, to answer.\n",
    "\n",
    "It's easy enough to *define* a [recursive](https://en.wikipedia.org/wiki/Recursion_(computer_science%29) function to calculate the chain length for a starting number $n$.\n",
    "$$\n",
    "f(n) = \\begin{cases}\n",
    "1 & n = 1 \\\\\n",
    "1+f(n/2) & n \\equiv 0 \\pmod{2} \\\\\n",
    "1+f(3n+1) & n \\neq 1\\ \\text{and}\\ n \\equiv 1 \\pmod{2}\n",
    "\\end{cases}\n",
    "$$\n",
    "However, we want its *implementation* to be efficient. We can optimize greatly if we cache the outputs we compute (the computer science term for this is [memoization](https://en.wikipedia.org/wiki/Memoization)). For instance, if we store the fact that $f(4) = 3$ after we've computed it, when we later compute $f(8) = 1 + f(4)$, the program can use the stored value of 3 rather than recomputing $f(4)$. For large inputs, this will save us (or really the computer, I guess) from redoing work.\n",
    "\n",
    "Python has a nice decorator called [cache](https://docs.python.org/3/library/functools.html) that will automagically memoize our function."
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    "from functools import cache\n",
    "\n",
    "@cache\n",
    "def collatz_length(n):\n",
    "    if n == 1:\n",
    "        return 1\n",
    "    elif n % 2 == 0:\n",
    "        return 1 + collatz_length(n // 2)\n",
    "    else:\n",
    "        return 1 + collatz_length(3 * n + 1)"
   ]
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       "837799"
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   "source": [
    "max(range(1, 1000000), key=collatz_length)"
   ]
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   "source": [
    "## Relevant sequences\n",
    "* Collatz chain lengths: [A008908](https://oeis.org/A008908)\n",
    "\n",
    "#### Copyright (C) 2025 filifa\n",
    "\n",
    "This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
   ]
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