Reciprocal cycles II
Problem 417
A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given:
| 1/2 | = | 0.5 |
| 1/3 | = | 0.(3) |
| 1/4 | = | 0.25 |
| 1/5 | = | 0.2 |
| 1/6 | = | 0.1(6) |
| 1/7 | = | 0.(142857) |
| 1/8 | = | 0.125 |
| 1/9 | = | 0.(1) |
| 1/10 | = | 0.1 |
Where 0.1(6) means 0.166666..., and has a 1-digit recurring cycle. It can be seen that 1/7 has a 6-digit recurring cycle.
Unit fractions whose denominator has no other prime factors than 2 and/or 5 are not considered to have a recurring cycle.
We define the length of the recurring cycle of those unit fractions as 0.
We define the length of the recurring cycle of those unit fractions as 0.
Let L(n) denote the length of the recurring cycle of 1/n.You are given that ∑ L(n) for 3 ≤ n ≤ 1 000 000 equals 55535191115.
Find ∑ L(n) for 3 ≤ n ≤ 100 000 000.