OEIS A000189: Number of solutions to x^3 ≡ 0 mod n

In this episode we unpack A000189, the count of residues x modulo n whose cube is 0 mod n. We reveal the multiplicative structure: for n = ∏ p^{e_p}, a(n) = ∏ p^{⌊2e_p/3⌋}. We'll illustrate with small n (a4 = 2, a9 = 3) and explain the equivalent form a(n) = ∏ p^{e_p − ⌈e_p/3⌉}, i.e., how many x in 0..n−1 satisfy p^{e_p} | x^3 across the prime powers. We also touch on the idea that a(n) equals the product of the corresponding prime-power counts, and how the largest cube-free divisor of n ties into the structure of cubes modulo n. A compact window into cubes modulo n and the power of multiplicativity in number theory.

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