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In this note we continue our study started in \\cite{Ul} of the arithmetic properties of the sequence of Stern polynomials and the sequence $\\{e(n)\\}_{n=1}^{\\infty}$. We also study the sequence $d(n)=\\op{ord}_{t=0}B_{n}(t)$. Among other things we prove that $d(n)=\\nu(n)$, where $\\nu(n)$ is the maximal power of 2 which dividies the number $n$. We also count the number of the solutions of the equations $e(m)=i$ and $e(m)-d(m)=i$ in the interval $[1,2^{n}]$. 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