Problem 346
Strong Repunits
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.
强循环单位数
7是一个特别的数，因为7在2进制下写作111，在6进制下写作11（也即710 = 116 = 1112）。换句话说，7在至少两种b > 1进制下为循环单位数。
我们称拥有这种性质的正整数为强循环单位数。可以验证，有8个小于50的强循环单位数：{1,7,13,15,21,31,40,43}。
进一步地，所有小于1000的强循环单位数之和是15864。
求所有小于1012的强循环单位数之和。