OEIS A000221: The happy numbers orbit of 5

In this Deep Dive we follow 5 under the map that replaces a number by the sum of the squares of its decimal digits. The sequence goes 5, 25, 29, 85, 89, and then falls into an eight-term cycle: 89, 145, 42, 24, 16, 37, 58, and back to 89. It never reaches 1, so 5 is not a happy number; it cycles instead. We discuss how this relates to the OEIS entry A000221, its connections to A0003132 and A00079, and notes from Hasler and Porges. The episode also highlights code samples in Mathematica, Magma, and Haskell that generate and visualize the iterates, illustrating how simple functions can exhibit rich structure.

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