12. The set of positive integers $n$ for which $\left[\log _{2} 1\right]+\left[\log _{2} 2\right]+\left[\log _{2} 3\right]+\cdots+\left[\log _{2} n\right]=2006$ holds is $\qquad$ .
12. $\varnothing\left[\log _{2} 1\right]+\left[\log _{2} 2\right]+\cdots+\left[\log _{2} 128\right]+\cdots+\left[\log _{2} 2^{55}\right]=2 \times 1+4 \times 2+8 \times 3+$ $16 \times 4+32 \times 5+64 \times 6+128 \times 7=1538$, so $\left[\log _{2} 256\right]+\left[\log _{2} 257\right]+\cdots+\left[\log _{2} n\right]=2006$ $-1538=468<256 \times 8$, and $8 \times 468$, hence there does not exist a satisfying $n$.