97 lines
2.6 KiB
Plaintext
97 lines
2.6 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "22f94a3b",
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"metadata": {},
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"source": [
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"# [Non-Abundant Sums](https://projecteuler.net/problem=23)\n",
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"\n",
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"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)."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "84f3ad0c",
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"metadata": {},
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"outputs": [],
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"source": [
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"aliquot_sum = lambda n: sigma(n) - n\n",
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"abundant_numbers = {k for k in range(1, 28124) if aliquot_sum(k) > k}"
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]
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},
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{
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"cell_type": "markdown",
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"id": "d6586d17",
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"metadata": {},
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"source": [
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"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."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "6747d2a3",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"4179871"
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]
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},
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"non_abundant_sums = set(range(1, 28124))\n",
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"for n in range(1, 28124):\n",
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" for m in abundant_numbers:\n",
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" if n - m in abundant_numbers:\n",
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" non_abundant_sums.discard(n)\n",
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" break\n",
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"\n",
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"sum(non_abundant_sums)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "91a43bd1",
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"metadata": {},
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"source": [
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"## Relevant sequences\n",
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"* Sums of divisors: [A000203](https://oeis.org/A000203)\n",
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"* Numbers that are not the sum of two abundant numbers: [A048242](https://oeis.org/A048242)\n",
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"\n",
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"#### Copyright (C) 2025 filifa\n",
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"\n",
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"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)."
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "SageMath 9.5",
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"language": "sage",
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"name": "sagemath"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.2"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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