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Find near miss prime multiples.

Given a number $n \geq 3$ as input output the smallest number $k$ of modular residues of $k$ by the first $n$ primes is exactly $\{-1,0,1\}$.
That is there is a prime in the first $n$ primes that divides $k$, one that divides $k+1$ and one that divides $k-1$, *and* every prime in the first $n$ primes divides one of those three values.
~~For example if $n=5$, the first $3$ primes are $2, 3, 5, 7, 11$ the value must be at least $10$ since with anything smaller $11$ is an issue. So we start with $10$:~~
$$
10 \equiv 0 \mod 2
$$
$$
10 \equiv 1 \mod 3
$$
$$
10 \equiv 0 \mod 5
$$
$$
10 \equiv 3 \mod 7
$$
Since $10 \mod 7 \equiv 3$, $10$ is not a solution so we try the next number.
* $11 \equiv 4\mod 7$ so we try the next number.
* $12 \equiv 2\mod 5$, so we try the next number.
* $13 \equiv 3\mod 5$, so we try the next number.
* $14 \equiv 3\mod 11$, so we try the next number.
* $15 \equiv 4\mod 11$, so we try the next number.
* $16 \equiv 2\mod 7$, so we try the next number.
* $17 \equiv 2\mod 5$, so we try the next number.
* $18 \equiv 3\mod 5$, so we try the next number.
* $19 \equiv 5\mod 7$, so we try the next number.
* $20 \equiv 9\mod 11$, so we try the next number.
$$
21 \equiv 1 \mod 2
$$
$$
21 \equiv 0 \mod 3
$$
$$
21 \equiv 1 \mod 5
$$
$$
21 \equiv 0 \mod 7
$$
$$
21 \equiv -1 \mod 11
$$
21 satisfies the property so its the answer.
## Task
Take as input $n \geq 3$ and give as output the smallest number satisfying.
This is code-golf. The goal is to minimize the size of your source code as measured in bytes
## Test cases
```
3 -> 4
4 -> 6
5 -> 21
6 -> 155
7 -> 441
```

Given a number $n \geq 3$ as input output the smallest number $k$ of modular residues of $k$ by the first $n$ primes is exactly $\{-1,0,1\}$.
That is there is a prime in the first $n$ primes that divides $k$, one that divides $k+1$ and one that divides $k-1$, *and* every prime in the first $n$ primes divides one of those three values.
For example if $n=5$, the first $5$ primes are $2, 3, 5, 7, 11$ the value must be at least $10$ since with anything smaller $11$ is an issue. So we start with $10$:
$$
10 \equiv 0 \mod 2
$$
$$
10 \equiv 1 \mod 3
$$
$$
10 \equiv 0 \mod 5
$$
$$
10 \equiv 3 \mod 7
$$
Since $10 \mod 7 \equiv 3$, $10$ is not a solution so we try the next number.
* $11 \equiv 4\mod 7$ so we try the next number.
* $12 \equiv 2\mod 5$, so we try the next number.
* $13 \equiv 3\mod 5$, so we try the next number.
* $14 \equiv 3\mod 11$, so we try the next number.
* $15 \equiv 4\mod 11$, so we try the next number.
* $16 \equiv 2\mod 7$, so we try the next number.
* $17 \equiv 2\mod 5$, so we try the next number.
* $18 \equiv 3\mod 5$, so we try the next number.
* $19 \equiv 5\mod 7$, so we try the next number.
* $20 \equiv 9\mod 11$, so we try the next number.
$$
21 \equiv 1 \mod 2
$$
$$
21 \equiv 0 \mod 3
$$
$$
21 \equiv 1 \mod 5
$$
$$
21 \equiv 0 \mod 7
$$
$$
21 \equiv -1 \mod 11
$$
21 satisfies the property so its the answer.
## Task
Take as input $n \geq 3$ and give as output the smallest number satisfying.
This is code-golf. The goal is to minimize the size of your source code as measured in bytes
## Test cases
```
3 -> 4
4 -> 6
5 -> 21
6 -> 155
7 -> 441
```

code-golf number-theory

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