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Difference of Two Series

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November 13, 2005
number theoryprime factorization

Problem Statement

For positive integers nn, let τ(n)\tau (n) denote the number of positive integer divisors of nn, including 11 and nn. For example, τ(1)=1\tau (1)=1 and τ(6)=4\tau(6) =4. Define S(n)S(n) by S(n)=τ(1)+τ(2)+...+τ(n).S(n)=\tau(1)+ \tau(2) + ... + \tau(n). Let aa denote the number of positive integers n≤2005n \leq 2005 with S(n)S(n) odd, and let bb denote the number of positive integers n≤2005n \leq 2005 with S(n)S(n) even. Find ∣a−b∣|a-b|.
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