s = 0
unseen = 0
seen(v) = bit test(s, v)
see(v) = s = bit or(s, 2^v); while (seen(unseen), unseen++)

A352724(n) = { my (r=[], o=0); while (n, my (v=valuation(n+n%2, 2)); if (n%2, r=concat(r, (2^v-1)*2^o)); o+=v; n\=2^v); r }

compute(n) = {
	my (r=A352724(n));
	for (v=unseen, oo,
		if (!seen(v) && setintersect(r, A352724(v))==[],
			see(v);
			return (v);
		);
	);
}

for (n=0, 2^13, print (n " " v=compute(n)))

quit