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Problem 346


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。