OEIS A000091: A multiplicative sequence and the Legendre/Jacobi shortcut

We explore A000091, a multiplicative OEIS sequence defined by prime-factor rules: powers of 2 map to 2; the prime 3 maps to 2 but higher powers map to 0; and primes p > 3 map to 2 or 0 according to p mod 3, with a universal override: any number divisible by 9 yields 0. We discuss how these rules can be implemented procedurally in Maple and Mathematica, using Legendre and Jacobi symbols to shortcut remainder tests, enabling fast evaluation for large n. Finally we pull back to the bigger picture: links to number theory, cryptography, and even signal processing, illustrating how a tiny rule set can illuminate connections across math.

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