## Strong Repunits

### Problem 346

Published on Saturday, 3rd September 2011, 04:00 pm; Solved by 2057; Difficulty rating: 15%
The number 7 is special, because 7 is 111 written in base 2, and 11 written in base 6

(i.e. 7_{10} = 11_{6} = 111_{2}). In other words, 7 is a repunit in at least two bases b > 1.

We shall call a positive integer with this property a strong repunit. It can be verified that there are 8 strong repunits below 50: {1,7,13,15,21,31,40,43}.

Furthermore, the sum of all strong repunits below 1000 equals 15864.

^{12}.