\\ A374356 a(n) is the greatest fibbinary number f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).
b(n) = { my (v = 0, e, x, y, b); while (n, x = y = 0; e = valuation(n, 2); for (k = 0, oo, if (bittest(n, e+k), n -= b = 2^(e+k); [x, y] = [y + b, x], v += x; break; ); ); ); return (v); }

{
	mx = 2^13-1;

	vv = vector(mx, n, []);
	for (n = 1, mx,
		v = b(n);
		vv[v] = concat(vv[v], n);
	);

	a = vector(mx);
	for (n = 1, #vv,
		for (k = 1, #vv[n],
			a[vv[n][k]] = vv[n][#vv[n]+1-k];
		);
	);

	print ("0 0");
	for (n = 1, #a,
		print (n " " a[n]);
	);
}

quit