51 lines
1.4 KiB
Markdown
51 lines
1.4 KiB
Markdown
|
---
|
||
|
|
id: 5900f5411000cf542c510052
|
||
|
|
title: 'Problem 467: Superinteger'
|
||
|
|
challengeType: 5
|
||
|
|
forumTopicId: 302142
|
||
|
|
dashedName: problem-467-superinteger
|
||
|
|
---
|
||
|
|
|
||
|
|
# --description--
|
||
|
|
|
||
|
|
An integer s is called a superinteger of another integer n if the digits of n form a subsequence of the digits of s.
|
||
|
|
|
||
|
|
For example, 2718281828 is a superinteger of 18828, while 314159 is not a superinteger of 151.
|
||
|
|
|
||
|
|
Let p(n) be the nth prime number, and let c(n) be the nth composite number. For example, p(1) = 2, p(10) = 29, c(1) = 4 and c(10) = 18. {p(i) : i ≥ 1} = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...} {c(i) : i ≥ 1} = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ...}
|
||
|
|
|
||
|
|
Let PD the sequence of the digital roots of {p(i)} (CD is defined similarly for {c(i)}): PD = {2, 3, 5, 7, 2, 4, 8, 1, 5, 2, ...} CD = {4, 6, 8, 9, 1, 3, 5, 6, 7, 9, ...}
|
||
|
|
|
||
|
|
Let Pn be the integer formed by concatenating the first n elements of PD (Cn is defined similarly for CD). P10 = 2357248152 C10 = 4689135679
|
||
|
|
|
||
|
|
Let f(n) be the smallest positive integer that is a common superinteger of Pn and Cn. For example, f(10) = 2357246891352679, and f(100) mod 1 000 000 007 = 771661825.
|
||
|
|
|
||
|
|
Find f(10 000) mod 1 000 000 007.
|
||
|
|
|
||
|
|
# --hints--
|
||
|
|
|
||
|
|
`euler467()` should return 775181359.
|
||
|
|
|
||
|
|
```js
|
||
|
|
assert.strictEqual(euler467(), 775181359);
|
||
|
|
```
|
||
|
|
|
||
|
|
# --seed--
|
||
|
|
|
||
|
|
## --seed-contents--
|
||
|
|
|
||
|
|
```js
|
||
|
|
function euler467() {
|
||
|
|
|
||
|
|
return true;
|
||
|
|
}
|
||
|
|
|
||
|
|
euler467();
|
||
|
|
```
|
||
|
|
|
||
|
|
# --solutions--
|
||
|
|
|
||
|
|
```js
|
||
|
|
// solution required
|
||
|
|
```
|