OEIS A000252: Number of invertible 2x2 matrices modulo n

We dive into OEIS A000252, the sequence that counts the invertible 2x2 matrices over the ring Z/nZ (the order of GL2(Z/nZ)). The sequence is multiplicative and ties together several core ideas in algebra: the prime-power formulas, the compact general form a(n) = n^4 ∏_{p|n} (1 − 1/p)(1 − 1/p^2), and the connection to SL2(Z/nZ) since |GL2(Z/nZ)| = φ(n) · |SL2(Z/nZ)|. We also explore how a(n) counts automorphisms of C_n × C_n and why these counts sit at a crossroads of modular arithmetic and group theory. Along the way we’ll see the prime-case building blocks (a(p) = (p^2−1)(p^2−p)) and how they multiply to give the full picture, plus notable properties like divisibility by 48 for n > 2 and the brisk growth reflected in the initial terms.

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