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A sequence related to the sigma function

Source: 2024 Turkey TST P8

March 18, 2024
number theory

Problem Statement

For an integer nn, σ(n)\sigma(n) denotes the sum of postitive divisors of nn. A sequence of positive integers (ai)i=0∞(a_i)_{i=0}^{\infty} with a0=1a_0 =1 is defined as follows: For each n>1n>1, ana_n is the smallest integer greater than 11 that satisfies σ(a0a1…an−1)∣σ(a0a1…an).\sigma{(a_0a_1\dots a_{n-1})} \vert \sigma{(a_0a_1\dots a_{n})}. Determine the number of divisors of 202420242024^{2024} amongst the sequence.
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