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    "# [Spiral Primes](https://projecteuler.net/problem=58)\n",
    "\n",
    "It's the return of the Ulam spiral from [problem 28](https://projecteuler.net/problem=28) (this time we're going counter-clockwise, but that doesn't actually affect much).\n",
    "\n",
    "We can handle this problem with a couple of easy-to-derive formulas. First, for a spiral with side length $n$ (note that $n$ must be odd), the number of diagonal entries is $2n-1$. Furthermore, the outermost diagonal entries will be $n^2$, $n^2 - (n-1)$, $n^2 - 2(n-1)$, and $n^2 - 3(n-1)$.\n",
    "\n",
    "With these facts, we can just iterate over odd values of $n$ and calculate the four outermost diagonal entries. We'll keep a running total $p$ of how many primes we see and stop when $\\frac{p}{2n-1} < 0.1$."
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   "source": [
    "def diagonal(n, k): return n^2 - k*(n-1)"
   ]
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       "26241"
      ]
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   "source": [
    "from itertools import count\n",
    "\n",
    "p = 0\n",
    "for n in count(3, 2):\n",
    "    for k in range(0, 4):\n",
    "        if is_prime(diagonal(n, k)):\n",
    "            p += 1\n",
    "            \n",
    "    if p / (2*n - 1) < 0.1:\n",
    "        break\n",
    "\n",
    "n"
   ]
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   "source": [
    "## Relevant sequences\n",
    "* Numbers on diagonals: [A200975](https://oeis.org/A200975)\n",
    "* Primes at right-angle turns on the Ulam spiral: [A172979](https://oeis.org/A172979)\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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