It is known a recurrence relation for the Ramanujan’s tau-function involving the sum of divisors function𝝈(𝒏), whose solution gives a closed formula for 𝝉(𝒏) in terms of complete Bell polynomials, and a determinantal expression for 𝝈(𝒎) where participate the values 𝝉(𝒌).
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Sivaraman et al. (2024) studied this question.
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