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Reciprocal cycles II

Problem 417

Published on Saturday, 2nd March 2013, 04:00 pm; Solved by 384

A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given:

1/20.5
1/30.(3)
1/40.25
1/50.2
1/60.1(6)
1/70.(142857)
1/80.125
1/90.(1)
1/100.1

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.

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