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Strong Repunits

Problem 346

Published on Saturday, 3rd September 2011, 04:00 pm; Solved by 1304

The number 7 is special, because 7 is 111 written in base 2, and 11 written in base 6
(i.e. 710 = 116 = 1112). 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.

Find the sum of all strong repunits below 1012.