OEIS A000216: The eight-term cycle of the sum-of-squares-of-digits map

We explore how the simple rule “replace a number by the sum of the squares of its decimal digits” behaves when starting from 2. Following the path 2 → 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4, the sequence falls into a fixed eight-term cycle. We unpack the idea of offset in OEIS entries, how A000216 relates to the digit-sum rule and to A08079 (the cycle starting at 4), and how a compact, PRR-like notation encodes the cycle using modulo arithmetic. The episode also touches on related streams—the beginning 1 fixed point, cross-references to other sequences like A003132 and A000218 for different starting points, and the broader notion of “semi-happy” numbers in the literature. We’ll connect historical notes (Berger 1970, Porges 1945) to modern perspectives and emphasize how such a tiny rule reveals structured, emergent behavior in number theory.

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