OEIS A000190: Counts the number of solutions to x^4 ≡ 0 (mod n)

In this episode we explore the OEIS entry A000190, which assigns to every n the number of residues x modulo n for which x^4 ≡ 0 (mod n). The function is multiplicative, and for prime powers p^e it has the neat formula a(p^e) = p^{⌊3e/4⌋}, so that a(n) = ∏ p^{⌊3e/4⌋} over the prime powers p^e dividing n. We’ll see how this leads to efficient computation from the prime factorization and why the sequence is a fundamental example of multiplicativity in number theory. We also discuss the Dirichlet generating function (an Euler product that signals the multiplicative structure) and connections to the so‑called shadow transform of fourth powers (A000583). Practical code snippets in Mathematica and GP illustrate both brute-force verification and the multiplicative approach, underscoring how prime-factorization unlocks counting problems modulo n.

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