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Möbius function and intervals

Problem 464

Published on Sunday, 23rd March 2014, 01:00 am; Solved by 133

The Möbius function, denoted μ(n), is defined as:

  • μ(n) = (-1)ω(n) if n is squarefree (where ω(n) is the number of distinct prime factors of n)
  • μ(n) = 0 if n is not squarefree.

Let P(a,b) be the number of integers n in the interval [a,b] such that μ(n) = 1.
Let N(a,b) be the number of integers n in the interval [a,b] such that μ(n) = -1.
For example, P(2,10) = 2 and N(2,10) = 4.

Let C(n) be the number of integer pairs (a,b) such that:

  • 1 ≤ a ≤ b ≤ n,
  • 99·N(a,b) ≤ 100·P(a,b), and
  • 99·P(a,b) ≤ 100·N(a,b).

For example, C(10) = 13, C(500) = 16676 and C(10 000) = 20155319.

Find C(20 000 000).