Definitions

The common methods to generate consecutive composites are

$$\overbrace{(n+1)! + 2, \ (n+1)! + 3, \ \ldots, \ (n+1)! + (n+1)}^{\text{n composites}}$$

$$\overbrace{n!+2,n!+3,...,n!+n}^{\text{n-1 composites}}$$

$$\overbrace{n\#+2,n\#+3,...n\#+n}^{\text{n-1 composites (Primorials)}}$$

A less common method involves the Chinese Remainder Theorem (CRT).

For example, define a list of \$n\$ pairwise coprimes \$p_1, p_2, \ldots, p_n\$.

A set of integers can also be called coprime if its elements share no common positive factor except 1. A stronger condition on a set of integers is pairwise coprime, which means that \$a\$ and \$b\$ are coprime for every pair \$(a, b)\$ of different integers in the set. The set \$\{2, 3, 4\}\$ is coprime, but it is not pairwise coprime since 2 and 4 are not relatively prime.

Create a set of \$n\$ congruences

\begin{align*} x + 1 &\equiv 0 \pmod{p_1} \\ x + 2 &\equiv 0 \pmod{p_2} \\ &\vdots \\ x + n &\equiv 0 \pmod{p_n} \end{align*}

By CRT, there exists a unique solution \$x\$ which satisfies this sequence of \$n\$ consecutive composite numbers:

$$x + 1, x + 2, \ldots, x + n$$

A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself.

Challenge

Given pairwise coprime inputs, find the smallest range of consecutive composite numbers.

Input

A list of \$n\$ pairwise coprimes.

Output

A sequence of \$n\$ consecutive composite numbers.

Test Cases

These cases were derived from this test.

Added a JavaScript (V8) tool to validate if the input numbers are pairwise coprime.