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   "source": [
    "We'll use the dual algorithm explained in [Linear prime-number sieves: a family tree](https://www.sciencedirect.com/science/article/pii/0167642387900244) to generate the primes. The primes themselves will be stored inside a [bit array](https://en.wikipedia.org/wiki/Bit_array)."
   ]
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    {
     "data": {
      "text/plain": [
       "142913828922"
      ]
     },
     "execution_count": 7,
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    }
   ],
   "source": [
    "import array\n",
    "\n",
    "def primes_upto(N):\n",
    "    S = array.array('B')\n",
    "    S.extend(0 for x in range((N >> 3) + 1))\n",
    "    def set_bit(i):\n",
    "        S[i>>3] |= 1<<(i&7)\n",
    "    def is_bit_set(i):\n",
    "        return S[i>>3]&(1<<(i&7)) \n",
    "    \n",
    "    f = 2\n",
    "    while f*2 <= N:\n",
    "        p = 2\n",
    "        pOK = True\n",
    "        while pOK:\n",
    "            set_bit(f*p)\n",
    "            t = p\n",
    "            p += 1\n",
    "            while is_bit_set(p):\n",
    "                p += 1\n",
    "            pOK = f*p <= N and (f % t != 0)\n",
    "        f += 1\n",
    "        \n",
    "    for i in range(2, N+1):\n",
    "        if not is_bit_set(i):\n",
    "            yield i\n",
    "\n",
    "sum(primes_upto(2_000_000))\n"
   ]
  }
 ],
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  "title": "Summation of primes"
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