In an array of $$$N$$$ positive integers, a coloring is a way of assigning exactly one color to each element of the array.
A coloring will be called prime if there exists a color $$$c$$$ such that all numbers painted with color $$$c$$$ sum to a prime number.
An array will be called harmonic if all of its colorings are prime, and its elements are between $$$1$$$ and $$$10^8$$$.
Given an $$$N$$$, determine whether there exists a harmonic array of size $$$N$$$. If it exists, output such an array.
A line with an integer $$$N$$$ ($$$1 \leq N \leq 13$$$).
If there does not exist a harmonic array of size $$$N$$$, output a line with the character '*'.
Otherwise, output a line with the $$$N$$$ elements of the harmonic array $$$A_1, \ldots, A_N$$$ ($$$1 \le A_i\le 10^8$$$).
If there are multiple solutions, any of them will be accepted.
2
29 8
4
2 3 5 3
In the first example, if we paint $$$29$$$ and $$$8$$$ with the same color, the only color that exists will have sum $$$29+8=37$$$, which is prime. On the other hand, if we paint $$$29$$$ and $$$8$$$ with different colors, the color of $$$29$$$ will have sum $$$29$$$, which is prime.
In the second example, some colorings of this array are:
In the first two colorings, the numbers painted red (which have a circle) sum to a prime number.
In the last two colorings, the numbers painted green (which have a diamond) sum to a prime number.
It can be shown that in this array, any other coloring is also prime.
