OEIS A000113: Transformation Groups and Dedekind's Psi

We unpack A000113, which counts the number of distinct transformation groups of order n. We’ll explore how prime factorization, Dedekind’s psi function, and the exponents arising from the factorization feed into the terms, and what strong divisibility and multiplicativity reveal about the structure of transformation groups. We'll connect these abstract ideas to concrete intuition, walk through a worked example (n = 4), and discuss what these properties say about building larger groups from smaller, coprime-order pieces and about the sequence’s growth.

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