In my answer to a recent MSE question I mention the elliptic curve LMFDB 1728.n4 with an infinite order generator and integer points $(-1,\pm1)$. The elliptic divisibility sequence associated with the generator is defined by initial values and recursions
$$ a_0 = 0,\; a_1 = 1,\; a_2 = 2,\; a_3 = -21,\; a_4 = -284, \\ a_{-n} = -a_n,\;\;\; a_n = (4a_{n-1}a_{n-3} + 21a_{n-2}^2)/a_{n-4}, \\ a_{2n+1} = a_{n+2}a_n^3 - a_{n+1}^3a_{n-1},\\ a_{2n} = a_n(a_{n+2}a_{n-1}^2 - a_{n-2}a_{n+1}^2)/2. $$
The sequence $a_n$ can be computed with the PARI/GP code
a(n) = subst(elldivpol(ellinit([0, 2]), n), 'x, -1)/(2 - n%2);
print(vector(9, n, a(n)))
/* [1, 2, -21, -284, 6989, 553350, -18912521, -20779585336, 5506106985351] */
I conjecture that $2^k|a_n$ if and only if $2^k|n$ for all integers $n>0, k\ge 0.$ In other words, $\nu_2(n) = \nu_2(a_n)$ for the $2$-adic valuation. I tested the conjecture up to $n=999.$ I can prove this is true for all $n$ odd. Perhaps this can be extended to a proof by induction on the $2$-adic valuation. Can anyone supply either a complete proof or else a counterexample?