From d682ac4aeb097aee00ce1679dfb8e4e10620b80f Mon Sep 17 00:00:00 2001
From: filifa <filifa.breath652@8alias.com>
Date: Thu, 26 Jun 2025 20:18:42 -0400
Subject: [PATCH] sieve instead

---
 notebooks/problem0070.ipynb | 78 ++++++++++++++++++-------------------
 1 file changed, 37 insertions(+), 41 deletions(-)

diff --git a/notebooks/problem0070.ipynb b/notebooks/problem0070.ipynb
index 0cd34ed..5441447 100644
--- a/notebooks/problem0070.ipynb
+++ b/notebooks/problem0070.ipynb
@@ -32,39 +32,57 @@
     "As in [problem 69](https://projecteuler.net/problem=69),\n",
     "$$\\phi(n) = n\\prod_{p | n} \\left(1 - \\frac{1}{p}\\right)$$\n",
     "\n",
-    "Rather than calculate the totients of every single number up to $10^7$, we'll start with just the primes - for a prime $p$, $\\phi(p) = p-1$. We'll store these primes in a [min-heap](https://en.wikipedia.org/wiki/Heap_(data_structure)) ordered by $\\frac{p}{\\phi(p)}$."
+    "We can calculate the totients of the numbers up to $10^7$ using a very similar approach to the [sieve of Eratosthenes](https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes) for generating prime numbers (see [problem 10](https://projecteuler.net/problem=10)).\n",
+    "\n",
+    "We'll initialize a list of numbers from 0 to $10^7$."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "244e05bd",
+   "id": "524fca40",
    "metadata": {},
    "outputs": [],
    "source": [
-    "import heapq\n",
-    "\n",
     "limit = 10^7\n",
-    "\n",
-    "primes = prime_range(limit)\n",
-    "queue = [(p / (p - 1), (p, p - 1)) for p in primes]\n",
-    "heapq.heapify(queue)"
+    "totients = list(range(0, limit))"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "9a057da0",
+   "id": "88ef58dd",
    "metadata": {},
    "source": [
-    "Then we'll search to find the value $n$ with the smallest ratio that is also a digit permutation of its totient. We'll check composite values by pushing $np$ to the queue for each prime $p$.\n",
-    "\n",
-    "When we find a value that is a digit permutation of its totient, we'll know that it also has the smallest ratio and can stop."
+    "Iterating $n$ from 2 to $10^7$, if `totients[n] == n`, then $n$ is prime, and we'll update its totient and all its multiples using the above formula. If `totients[n] != n`, then we'll check if $n/\\phi(n)$ is small and if $\\phi(n)$ is a permutation of $n$, keeping track of the best answer so far."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "99c4267a",
+   "id": "aa071cc4",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "answer = None\n",
+    "ratio = float('inf')\n",
+    "\n",
+    "for n in range(2, limit):\n",
+    "    if totients[n] != n:\n",
+    "        r = n / totients[n]\n",
+    "        if r < ratio and is_permutation_pair(n, totients[n]):\n",
+    "            ratio = r\n",
+    "            answer = n\n",
+    "\n",
+    "        continue\n",
+    "\n",
+    "    for p in range(n, limit, n):\n",
+    "        totients[p] -= totients[p] // n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "d4819aa5",
    "metadata": {},
    "outputs": [
     {
@@ -73,38 +91,12 @@
        "8319823"
       ]
      },
-     "execution_count": 3,
+     "execution_count": 4,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "answer = None\n",
-    "visited = set()\n",
-    "while queue != []:\n",
-    "    _, (n, totient) = heapq.heappop(queue)\n",
-    "        \n",
-    "    if n in visited:\n",
-    "        continue\n",
-    "    visited.add(n)\n",
-    "        \n",
-    "    if is_permutation_pair(n, totient):\n",
-    "        answer = n\n",
-    "        break\n",
-    "        \n",
-    "    for p in primes:\n",
-    "        q = n * p\n",
-    "        if q >= limit:\n",
-    "            break\n",
-    "            \n",
-    "        if n % p != 0:\n",
-    "            new_totient = totient * (p - 1)\n",
-    "        else:\n",
-    "            new_totient = totient * p\n",
-    "        \n",
-    "        ratio = q / new_totient\n",
-    "        heapq.heappush(queue, (ratio, (q, new_totient)))\n",
-    "        \n",
     "answer"
    ]
   },
@@ -113,7 +105,11 @@
    "id": "40cf1e01",
    "metadata": {},
    "source": [
-    "Note: lots of people in the problem thread make the assumption that the answer must be a [semiprime](https://en.wikipedia.org/wiki/Semiprime). However, Steendor points out that for certain upper bounds, this assumption does not hold. This solution avoids making that assumption."
+    "Note: lots of people in the problem thread make the assumption that the answer must be a [semiprime](https://en.wikipedia.org/wiki/Semiprime). However, Steendor points out that for certain upper bounds, this assumption does not hold. For instance, $2817 = 3^2 \\times 313$ and $\\phi(2817) = 1872$, and 2817/1872 is the lowest ratio until 2991. 2817 may be an exception rather than the norm (all the other numbers in A102018 up to $10^8$ are semiprimes); nevertheless, this solution avoids making the assumption.\n",
+    "\n",
+    "## Relevant sequences\n",
+    "* All numbers $n$ such that $\\phi(n)$ is a digit permutation: [A115921](https://oeis.org/A115921)\n",
+    "* Subsequence of A115921 such that $n/\\phi(n)$ is a record low: [A102018](https://oeis.org/A102018)"
    ]
   }
  ],