MathDB
2021 USMCA National Championship #20

Source:

May 9, 2021

Problem Statement

Let τ(n)\tau(n) be the number of positive divisors of nn, let f(n)=∑d∣nτ(d)f(n) = \sum_{d \mid n} \tau(d), and let g(n)=∑d∣nf(d)g(n) = \sum_{d \mid n} f(d). Let PnP_n be the product of the first nn prime numbers, and let M=P1P2⋯P2021M = P_1 P_2 \cdots P_{2021}. Then ∑d∣M1g(d)=ab\sum_{d \mid M} \frac{1}{g(d)} = \frac{a}{b}, where a,ba, b are relatively prime positive integers. What is the remainder when τ(ab)\tau(ab) is divided by 20172017? (Here, ∑d∣n\sum_{d \mid n} means a sum over the positive divisors of nn.)
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