57 lines
1.6 KiB
Markdown
57 lines
1.6 KiB
Markdown
---
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id: 5900f4811000cf542c50ff94
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title: 'Problem 277: A Modified Collatz sequence'
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challengeType: 5
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forumTopicId: 301927
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dashedName: problem-277-a-modified-collatz-sequence
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---
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# --description--
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A modified Collatz sequence of integers is obtained from a starting value $a_1$ in the following way:
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$a_{n + 1} = \frac{a_n}{3}$ if $a_n$ is divisible by 3. We shall denote this as a large downward step, "D".
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$a_{n + 1} = \frac{4a_n + 2}{3}$ if $a_n$ divided by 3 gives a remainder of 1. We shall denote this as an upward step, "U".
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$a_{n + 1} = \frac{2a_n - 1}{3}$ if $a_n$ divided by 3 gives a remainder of 2. We shall denote this as a small downward step, "d".
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The sequence terminates when some $a_n = 1$.
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Given any integer, we can list out the sequence of steps. For instance if $a_1 = 231$, then the sequence $\\{a_n\\} = \\{231, 77, 51, 17, 11, 7, 10, 14, 9, 3, 1\\}$ corresponds to the steps "DdDddUUdDD".
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Of course, there are other sequences that begin with that same sequence "DdDddUUdDD....".
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For instance, if $a_1 = 1004064$, then the sequence is DdDddUUdDDDdUDUUUdDdUUDDDUdDD.
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In fact, 1004064 is the smallest possible $a_1 > {10}^6$ that begins with the sequence DdDddUUdDD.
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What is the smallest $a_1 > {10}^{15}$ that begins with the sequence "UDDDUdddDDUDDddDdDddDDUDDdUUDd"?
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# --hints--
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`modifiedCollatzSequence()` should return `1125977393124310`.
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```js
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assert.strictEqual(modifiedCollatzSequence(), 1125977393124310);
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```
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# --seed--
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## --seed-contents--
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```js
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function modifiedCollatzSequence() {
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return true;
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}
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modifiedCollatzSequence();
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```
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# --solutions--
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```js
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// solution required
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```
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