Prime factor count

PrimeFactorCount.ipynb

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   "cell_type": "code",
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   "source": [
    "# Use inline SVGs for plots\n",
    "%config InlineBackend.figure_formats = ['svg']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pylab as plt\n",
    "import seaborn as sns\n",
    "\n",
    "from functools import reduce"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "sns.set_style(\"whitegrid\")"
   ]
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  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "PLT_FIGSIZE = (14, 5.5)\n",
    "n = 2048"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# https://stackoverflow.com/a/6800214\n",
    "def factors(n):\n",
    "    return set(reduce(list.__add__,\n",
    "                ([i, n // i] for i in range(1, int(n ** 0.5) + 1) if n % i == 0)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = np.arange(1, n)\n",
    "y = np.array([len(factors(x_i)) for x_i in X])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Get first occurences of divisor count\n",
    "d, d_idx = np.unique(y, return_index=True)\n",
    "\n",
    "# Boolean indices for easier access\n",
    "d_idxb = np.isin(X, d_idx + 1, assume_unique=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Filter for monotonously increasing values by index\n",
    "d_prime = d_idx[1:] - d_idx[:-1]\n",
    "# d1_idx = d_idx[np.where(d_prime >= 0)]\n",
    "d2_idx = d_idx[np.where(d_prime < 0)]\n",
    "d1_idx = np.setdiff1d(d_idx, d2_idx)\n",
    "\n",
    "# Boolean indices for easier access\n",
    "d1_idxb = np.isin(X, d1_idx + 1, assume_unique=True)\n",
    "d2_idxb = np.isin(X, d2_idx + 1, assume_unique=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Linear/logarithmic fit\n",
    "from scipy.optimize import curve_fit\n",
    "\n",
    "def func(x, a, b, c, d):\n",
    "    # return a * np.exp(-b * x) + c\n",
    "    return a * np.log(b * x) + c * x + d\n",
    "\n",
    "popt, pcov = curve_fit(func, X[d1_idxb], y[d1_idxb], bounds=(0, np.infty))"
   ]
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   "cell_type": "code",
   "execution_count": 10,
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      "text/plain": [
       "<Figure size 1008x396 with 1 Axes>"
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   ],
   "source": [
    "# Plot data and fit\n",
    "fig = plt.figure(figsize=PLT_FIGSIZE)\n",
    "plt.scatter(X[d1_idxb], y[d1_idxb], c='C0', s=24, marker='x',\n",
    "            label='First occurences (monotonously increasing indices)')\n",
    "plt.scatter(X[d2_idxb], y[d2_idxb], c='C3', s=24, marker='x',\n",
    "            label='First occurences (other)')\n",
    "plt.scatter(X[~d_idxb], y[~d_idxb], c='C1', s=3,\n",
    "            label='Later occurences')\n",
    "plt.plot(X, func(X, *popt), 'r-', c='C2',\n",
    "         label='Linear/Logarithmic fit')\n",
    "plt.xlabel('$n \\in \\mathbb{N}$')\n",
    "plt.ylabel('Prime factor count')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
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