We dive into A000351, the powers of five, from its tidy recurrence and generating function to the surprising ways it shows up across number theory, combinatorics, and geometry. Along the way we explore connections to Pisot-number phenomena, a divisor-sum–based lens on primality, counting integers with only odd digits, generating-function identities, and even a fractal-counting angle in the Sierpinski pyramid. We also glimpse neat infinite-series quirks: the sum of reciprocals of 5^n equals 5/4 and the alternating reciprocal sum equals 5/6. Finally, we pose a thought-provoking parallel: what would a similar sigma-based lens reveal if we looked at A000244, the powers of three? A compact seed that grows into a network of ideas.

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