A NOTE ON GAPS BETWEEN HAPPY NUMBERS

Abstract

Fix a base $b\ge 2$ and an exponent $e\ge 2$. An $e$-power $b$-happy number is a positive integer that reaches $1$ under iteration of the function mapping a positive integer to the sum of the $e$-th powers of its base $b$ digits. In this note, we answer the question of how large the gaps between $e$-power $b$-happy numbers can be.