Between the Squares: Primes, Floor Roots, and Legendre's Conjecture

We dive into an OEIS sequence built from primes by taking the integer part of their square roots and interpret it as a guide to how many perfect squares lie beneath each prime. Along the way we’ll see why this pattern can only creep up by at most one step at a time, connect it to Legendre’s famous conjecture about primes between consecutive squares, and explore what these frequencies of values reveal about how primes are spread out. It’s a math detective story that links number theory, pattern hunting, and real-world ideas like cryptography—all by looking at the simple act of squaring and taking roots.

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