55 lines
1.5 KiB
Markdown
55 lines
1.5 KiB
Markdown
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---
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id: 5900f4e51000cf542c50fff6
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title: 'Problem 374: Maximum Integer Partition Product'
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challengeType: 5
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forumTopicId: 302036
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dashedName: problem-374-maximum-integer-partition-product
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---
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# --description--
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An integer partition of a number $n$ is a way of writing $n$ as a sum of positive integers.
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Partitions that differ only in the order of their summands are considered the same. A partition of $n$ into distinct parts is a partition of $n$ in which every part occurs at most once.
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The partitions of 5 into distinct parts are:
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5, 4 + 1 and 3 + 2.
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Let $f(n)$ be the maximum product of the parts of any such partition of $n$ into distinct parts and let $m(n)$ be the number of elements of any such partition of $n$ with that product.
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So $f(5) = 6$ and $m(5) = 2$.
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For $n = 10$ the partition with the largest product is $10 = 2 + 3 + 5$, which gives $f(10) = 30$ and $m(10) = 3$. And their product, $f(10) \times m(10) = 30 \times 3 = 90$
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It can be verified that $\sum f(n) \times m(n)$ for $1 ≤ n ≤ 100 = 1\\,683\\,550\\,844\\,462$.
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Find $\sum f(n) \times m(n)$ for $1 ≤ n ≤ {10}^{14}$. Give your answer modulo $982\\,451\\,653$, the 50 millionth prime.
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# --hints--
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`maximumIntegerPartitionProduct()` should return `334420941`.
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```js
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assert.strictEqual(maximumIntegerPartitionProduct(), 334420941);
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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 maximumIntegerPartitionProduct() {
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return true;
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}
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maximumIntegerPartitionProduct();
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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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