From 8d954da214687718ec9173eb82be96095e77954b Mon Sep 17 00:00:00 2001
From: filifa <filifa.breath652@8alias.com>
Date: Thu, 19 Jun 2025 00:19:18 -0400
Subject: [PATCH] add problem 95

---
 notebooks/problem0095.ipynb | 110 ++++++++++++++++++++++++++++++++++++
 1 file changed, 110 insertions(+)
 create mode 100644 notebooks/problem0095.ipynb

diff --git a/notebooks/problem0095.ipynb b/notebooks/problem0095.ipynb
new file mode 100644
index 0000000..d604a34
--- /dev/null
+++ b/notebooks/problem0095.ipynb
@@ -0,0 +1,110 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "a84baacf",
+   "metadata": {},
+   "source": [
+    "# [Amicable Chains](https://projecteuler.net/problem=95)\n",
+    "\n",
+    "Numbers that form an amicable chain are called [sociable numbers](https://en.wikipedia.org/wiki/Sociable_number). Interestingly, the [Catalan-Dickson conjecture](https://en.wikipedia.org/wiki/Aliquot_sequence) posits that *every* starting number eventually reaches either 0 or a sociable number.\n",
+    "\n",
+    "Regardless, we can find these chains very cleanly, albeit somewhat slowly, with SageMath's graph tooling and `divisors` function. Simply add a directed edge from every number to its aliquot sum, assuming both are below 1000000."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "f047494c",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Looped digraph on 965607 vertices (use the .plot() method to plot)"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "limit = 1000000\n",
+    "G = DiGraph(loops=True)\n",
+    "for n in range(1, limit + 1):\n",
+    "    s = sum(divisors(n)) - n\n",
+    "    if s <= limit:\n",
+    "        G.add_edge(n, s)\n",
+    "        \n",
+    "G"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "38ac2d0e",
+   "metadata": {},
+   "source": [
+    "Once the graph is constructed, just iterate through all the cycles to get to the largest one, and get its smallest number."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "8527e13c",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "14316"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "longest_cycle = None\n",
+    "for cycle in G.all_cycles_iterator(simple=True):\n",
+    "    longest_cycle = cycle\n",
+    "    \n",
+    "min(longest_cycle)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9bf72629",
+   "metadata": {},
+   "source": [
+    "Another interesting fact: the amicable chain we've found here is not just the longest chain composed of numbers below 1000000 - it's actually the longest known amicable chain, *period*.\n",
+    "\n",
+    "## Relevant sequences\n",
+    "* Smallest members of amicable chains: [A003416](https://oeis.org/A003416)\n",
+    "* The amicable chain containing this problem's answer: [A072890](https://oeis.org/A072890)"
+   ]
+  }
+ ],
+ "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
+}