76 lines
2.5 KiB
Plaintext
76 lines
2.5 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "b6aef7e5",
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"metadata": {},
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"source": [
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"# [Counting Fractions](https://projecteuler.net/problem=72)\n",
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"\n",
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"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",
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"\n",
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"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",
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"\n",
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"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",
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"$$|F_{n+1}| = |F_n| + \\phi(n+1)$$\n",
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"From this, we can derive a non-recursive formula:\n",
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"$$|F_n| = 1 + \\sum_{k=1}^n \\phi(k)$$\n",
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"\n",
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"As mentioned before, we'll actually subtract two from this total, since the problem isn't counting 0 or 1."
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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": "2d21b0a4",
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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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"303963552391"
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]
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},
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"execution_count": 1,
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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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"sum(euler_phi(n) for n in range(1, 1000001)) - 1"
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]
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},
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{
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"cell_type": "markdown",
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"id": "1c67d8da",
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"metadata": {},
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"source": [
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"## Relevant sequences\n",
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"* Cardinalities of Farey sequences: [A005728](https://oeis.org/A005728)\n",
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"* Partial sums of totient function: [A002088](https://oeis.org/A002088)"
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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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