OEIS A000095: Fixed points of Γ0(N) in the modular group

A deep dive into A000095, the multiplicative sequence counting fixed points of the modular subgroup Γ0(N) on the upper half-plane. We explore how these fixed points connect modular transformations, Legendre/Jacobi symbols, and prime factorization, with examples like A2 = 2 and A_p = 2 for primes p ≡ 1 mod 4. We discuss the computation via code (Maple and Mathematica snippets), the surprising appearance of the constant 2π in its asymptotic mean, and the broader questions and potential applications in number theory.

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