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A perfect polynomial over ${𝔽}_2$ is a polynomial $A\in {𝔽}_2\left[x\right]$ that equals the sum of all its divisors. If $gcd(A,x^2+x)=1$ then we say that $A$ is odd. In this paper we show the non-existence of odd perfect polynomials with either three prime divisors or with at most nine prime divisors provided that all exponents are equal to $2.$

In this paper, we study the properties of the sequence of polynomials given by $g_0=0,g_1=1$, $g_{n+1}=g_n+\Delta g_{n-1}$ for $n\ge 1$, where $\Delta \in {𝔽}_q\left[t\right]$ is non-constant and the characteristic of ${𝔽}_q$ is $2$. This complements some results from R. Euler, L.H. Gallardo: On explicit formulae and linear recurrent sequences, Acta Math. Univ. Comenianae, 80 (2011) 213-219.