Puzzle 1166 Two puzzles from Mr. N. Nomoto-II

Puzzle 1166 Two puzzles from Mr. N. Nomoto-II
Mr. Naohiro Nomoto sent two puzzles. In this page I will publish only the second one.

Product of P_{n}(x) = ( A002110(n)/2 )^4　.

x is a set of distinct even numbers.

n = 6,　x = 2,4,6,8,10.

n = 8, x = 2,4,6,14,16.

n = 9, x = 4,8,10,12,14,18.

n = 11, x = 6,8,14,16,22,24,28.

n = 13, x = 6,14.16.22.26.28.30.36.

n = 14, x = 6,12,16,28,32,34,38.

n = 15, x = 6.12.16,32,34,38,44.

The product of P_{n}(x) is equal to A002110(n)/2 to the fourth power.

Q. Is it possible for the product to reach the fourth power even when n is larger than 15?