OEIS A000218: Sum of squares of digits

We explore the classic A000218, the rule that replaces a number with the sum of the squares of its decimal digits. Starting from 3, the path is 3 → 9 → 81 → 65 → 61 → 37 and then settles into the eight-term cycle 37 → 58 → 89 → 145 → 42 → 20 → 4 → 16 → 37. We’ll connect this to happy and unhappy numbers, Porgis’s 1945 result, and generalizations to sums of other powers and to different bases.

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