2018-09-30 22:01:58 +00:00
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---
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id: 5900f5131000cf542c510024
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title: 'Problem 421: Prime factors of n15+1'
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2020-11-27 18:02:05 +00:00
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challengeType: 5
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2019-08-05 16:17:33 +00:00
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forumTopicId: 302091
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2021-01-13 02:31:00 +00:00
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dashedName: problem-421-prime-factors-of-n151
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2018-09-30 22:01:58 +00:00
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---
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2020-11-27 18:02:05 +00:00
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# --description--
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2018-09-30 22:01:58 +00:00
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2021-07-29 18:14:09 +00:00
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Numbers of the form $n^{15} + 1$ are composite for every integer $n > 1$.
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2018-09-30 22:01:58 +00:00
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2021-07-29 18:14:09 +00:00
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For positive integers $n$ and $m$ let $s(n, m)$ be defined as the sum of the distinct prime factors of $n^{15} + 1$ not exceeding $m$.
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2018-09-30 22:01:58 +00:00
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2021-07-29 18:14:09 +00:00
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E.g. $2^{15} + 1 = 3 × 3 × 11 × 331$.
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2018-09-30 22:01:58 +00:00
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2021-07-29 18:14:09 +00:00
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So $s(2, 10) = 3$ and $s(2, 1000) = 3 + 11 + 331 = 345$.
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Also ${10}^{15} + 1 = 7 × 11 × 13 × 211 × 241 × 2161 × 9091$.
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So $s(10, 100) = 31$ and $s(10, 1000) = 483$.
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Find $\sum s(n, {10}^8)$ for $1 ≤ n ≤ {10}^{11}$.
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2018-09-30 22:01:58 +00:00
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2020-11-27 18:02:05 +00:00
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# --hints--
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2018-09-30 22:01:58 +00:00
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2021-07-29 18:14:09 +00:00
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`primeFactorsOfN15Plus1()` should return `2304215802083466200`.
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2018-09-30 22:01:58 +00:00
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2020-11-27 18:02:05 +00:00
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```js
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2021-07-29 18:14:09 +00:00
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assert.strictEqual(primeFactorsOfN15Plus1(), 2304215802083466200);
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2018-09-30 22:01:58 +00:00
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```
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2020-11-27 18:02:05 +00:00
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# --seed--
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2018-09-30 22:01:58 +00:00
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2020-11-27 18:02:05 +00:00
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## --seed-contents--
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2018-09-30 22:01:58 +00:00
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```js
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2021-07-29 18:14:09 +00:00
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function primeFactorsOfN15Plus1() {
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2020-09-15 16:57:40 +00:00
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2018-09-30 22:01:58 +00:00
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return true;
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}
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2021-07-29 18:14:09 +00:00
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primeFactorsOfN15Plus1();
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2018-09-30 22:01:58 +00:00
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```
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2020-11-27 18:02:05 +00:00
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# --solutions--
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2018-09-30 22:01:58 +00:00
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```js
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// solution required
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```
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