Intellectually Curious
Intellectually Curious is a podcast by Mike Breault featuring over 1,200 AI-powered explorations across science, mathematics, philosophy, and personal growth. Each short-form episode is generated, refined, and published with the help of large language models—turning curiosity into an ongoing audio encyclopedia. Designed for anyone who loves learning, it offers quick dives into everything from combinatorics and cryptography to systems thinking and psychology.
Inspiration for this podcast:
“Muad'Dib learned rapidly because his first training was in how to learn. And the first lesson of all was the basic trust that he could learn. It's shocking to find how many people do not believe they can learn, and how many more believe learning to be difficult. Muad'Dib knew that every experience carries its lesson.”
― Frank Herbert, Dune
Note: These podcasts were made with NotebookLM. AI can make mistakes. Please double-check any critical information.
Intellectually Curious
OEIS A000330: Square pyramidal numbers
In this episode we dive into A000330, the square pyramidal numbers, defined by a(n) = 0^2 + 1^2 + 2^2 + ... + n^2 = n*(n+1)*(2*n+1)/6. We’ll see why these count cannonball pyramids with square bases and, in the 2D analogue, the total number of squares in an n×n grid. We discuss the key identity S(n) = T(n) + T(n−1), where T(k) are tetrahedral numbers, linking square pyramidal numbers to other figurate families. We’ll cover famous results: the only square pyramidal number greater than 1 that is also a perfect square is 4900, and no square pyramidal number greater than 1 is tetrahedral. We’ll also explore interesting number-theoretic properties—units digits form a period-20 cycle, and n divides S(n) iff n ≡ ±1 (mod 6). Finally, we glimpse a tantalizing conjecture that every integer can be expressed as a sum of three generalized square pyramidal numbers. A rich tour of geometry, combinatorics, and modular arithmetic awaits.
Note: This podcast was AI-generated, and sometimes AI can make mistakes. Please double-check any critical information.
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