eulerbooks/notebooks/problem0023.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "22f94a3b",
   "metadata": {},
   "source": [
    "# [Non-Abundant Sums](https://projecteuler.net/problem=23)\n",
    "\n",
    "Just like in [problem 21](https://projecteuler.net/problem=21), we'll define an `aliquot_sum` function and use that to find all the [abundant numbers](https://en.wikipedia.org/wiki/Abundant_number) below 28,124 (as in problem 21, we could instead use a sieve to compute the divisor sums)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "84f3ad0c",
   "metadata": {},
   "outputs": [],
   "source": [
    "aliquot_sum = lambda n: sigma(n) - n\n",
    "abundant_numbers = {k for k in range(1, 28124) if aliquot_sum(k) > k}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d6586d17",
   "metadata": {},
   "source": [
    "Then we check every integer less than 28,124 to see if it's the sum of any two abundant numbers, and if it is, remove it from a set containing all those integers. Whatever's left in that set are the non-abundant sums."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "6747d2a3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4179871"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "non_abundant_sums = set(range(1, 28124))\n",
    "for n in range(1, 28124):\n",
    "    for m in abundant_numbers:\n",
    "        if n - m in abundant_numbers:\n",
    "            non_abundant_sums.discard(n)\n",
    "            break\n",
    "\n",
    "sum(non_abundant_sums)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "91a43bd1",
   "metadata": {},
   "source": [
    "## Relevant sequences\n",
    "* Sums of divisors: [A000203](https://oeis.org/A000203)\n",
    "* Numbers that are not the sum of two abundant numbers: [A048242](https://oeis.org/A048242)\n",
    "\n",
    "#### Copyright (C) 2025 filifa\n",
    "\n",
    "This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
   ]
  }
 ],
 "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
}