Is the set of polynomials an $x^n+ a_{n-1}x^{n-1}+\ldots+a_1x +a_0$ such that $2^k+1$ divides $a_k$ an ideal in $\Bbb Z[x]$?

Question

Is the set of polynomials $a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0$ such that $2^k+1$ divides $a_k$ an ideal in $\Bbb Z[x]$?

I think it is true for $2^k+1$ and it will be true for all the divisors as they will be taken common when trying to prove closure under addition and multiplication while proving for ideals

Answer 1

$2$ is in the set, but $2x$ isn't.