add problem 72

notebooks/problem0072.ipynb

@@ -0,0 +1,75 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "b6aef7e5",
+   "metadata": {},
+   "source": [
+    "# [Counting Fractions](https://projecteuler.net/problem=72)\n",
+    "\n",
+    "Like [problem 71](https://projecteuler.net/problem=71), we're looking at [Farey sequences](https://en.wikipedia.org/wiki/Farey_sequence). This time we're interested in the cardinality of $F_{1000000}$.\n",
+    "\n",
+    "To begin, first note that $F_1 = \\{0, 1\\}$, so $|F_1| = 2$ (this problem isn't counting 0 and 1 in its totals - we'll handle that at the end). Then consider that for any Farey sequence $F_n$, the next sequence $F_{n+1}$ will contain all the terms from $F_n$, along with all irreducible fractions $\\frac{k}{n+1}$, (since any *reducible* fraction would already be in $F_n$).\n",
+    "\n",
+    "How many new fractions does this get us? Well, the fraction only reduces if $k$ and $n+1$ have a common factor - in other words, if $k$ and $n+1$ are coprime, the fraction will not reduce. How many number less than $n+1$ are coprime to $n+1$? The [totient function](https://en.wikipedia.org/wiki/Euler%27s_totient_function) will tell us! So the number of irreducible fractions with denominator $n+1$ is simply $\\phi(n+1)$ This gives us\n",
+    "$$|F_{n+1}| = |F_n| + \\phi(n+1)$$\n",
+    "From this, we can derive a non-recursive formula:\n",
+    "$$|F_n| = 1 + \\sum_{k=1}^n \\phi(k)$$\n",
+    "\n",
+    "As mentioned before, we'll actually subtract two from this total, since the problem isn't counting 0 or 1."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "2d21b0a4",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "303963552391"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "sum(euler_phi(n) for n in range(1, 1000001)) - 1"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1c67d8da",
+   "metadata": {},
+   "source": [
+    "## Relevant sequences\n",
+    "* Cardinalities of Farey sequences: [A005728](https://oeis.org/A005728)\n",
+    "* Partial sums of totient function: [A002088](https://oeis.org/A002088)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "SageMath 9.5",
+   "language": "sage",
+   "name": "sagemath"
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+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
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+   "file_extension": ".py",
+   "mimetype": "text/x-python",
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+   "nbconvert_exporter": "python",
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+}