add problem 77

notebooks/problem0077.ipynb

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "5411064f",
+   "metadata": {},
+   "source": [
+    "# [Prime Summations](https://projecteuler.net/problem=77)\n",
+    "\n",
+    "Once again, we can adapt our solution to [problem 31](https://projecteuler.net/problem=31). Here, there's the added wrinkle that we don't know how far out we need to calculate our generating function, but we can work around this by just repeatedly increasing our precision and recalculating until we find our answer."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "1b5376e2",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "1 + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 4*x^9 + 5*x^10 + 6*x^11 + 7*x^12 + 9*x^13 + 10*x^14 + 12*x^15 + 14*x^16 + 17*x^17 + 19*x^18 + 23*x^19 + 26*x^20 + 30*x^21 + 35*x^22 + 40*x^23 + 46*x^24 + 52*x^25 + 60*x^26 + 67*x^27 + 77*x^28 + 87*x^29 + 98*x^30 + 111*x^31 + 124*x^32 + 140*x^33 + 157*x^34 + 175*x^35 + 197*x^36 + 219*x^37 + 244*x^38 + 272*x^39 + 302*x^40 + 336*x^41 + 372*x^42 + 413*x^43 + 456*x^44 + 504*x^45 + 557*x^46 + 614*x^47 + 677*x^48 + 744*x^49 + 819*x^50 + 899*x^51 + 987*x^52 + 1083*x^53 + 1186*x^54 + 1298*x^55 + 1420*x^56 + 1552*x^57 + 1695*x^58 + 1850*x^59 + 2018*x^60 + 2198*x^61 + 2394*x^62 + 2605*x^63 + 2833*x^64 + 3079*x^65 + 3344*x^66 + 3630*x^67 + 3936*x^68 + 4268*x^69 + 4624*x^70 + 5007*x^71 + 5419*x^72 + 5861*x^73 + 6336*x^74 + 6845*x^75 + 7393*x^76 + 7979*x^77 + 8608*x^78 + 9282*x^79 + O(x^80)"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "prec = 20\n",
+    "while True:\n",
+    "    R.<x> = PowerSeriesRing(ZZ, default_prec=prec)\n",
+    "    G = 1 / prod(1 - x^p for p in primes_first_n(prec))\n",
+    "    \n",
+    "    d = G.dict()\n",
+    "    if any(c > 5000 for c in d.values()):\n",
+    "        break\n",
+    "        \n",
+    "    prec *= 2\n",
+    "\n",
+    "G"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "af318759",
+   "metadata": {},
+   "source": [
+    "We can see that $x^{71}$ is our first term with a coeffcient over 5000, so our answer is 71.\n",
+    "\n",
+    "## Relevant sequences\n",
+    "* Number of partitions of $n$ into prime parts: [A000607](https://oeis.org/A000607)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "SageMath 9.5",
+   "language": "sage",
+   "name": "sagemath"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}