https://oeis.org/A305942 I recently extended this conjectured sequence (the number of decimal expansions of powers of two that contain exactly n digits 0). There would of course be analog sequences for the other nine digits. I wondered if any of these ten sequences might have 0 as a term. In other words, is there an n for which the decimal digit d does not appear in any power of two exactly n times? The digit 7 appears exactly 275923 times for *only eight* powers of two, the smallest number of solutions that I found (for any d, n up to 295000). It would take a gargantuan effort to bring this down to seven or six solutions, so no, I don't think zero is ever going to be a known. Against a backdrop of all counts of the digit 7 in powers of two from 2^9100000 to 2^9240000, here are my eight solutions in graphical form: http://chesswanks.com/num/eight.png