OEIS A000021: Quadratic forms, primes, and the count of representable integers

Dive with us into OEIS A000021, the sequence that counts how many integers up to 2^n can be written in the form x^2 + 12y^2. We’ll unpack this quadratic form as a geometric blueprint, explore its ties to Landau’s ideas about primes, and connect it to Jacobi’s four-square theorem. Along the way we’ll glimpse the web of related sequences (like R000021) and meet the mathematicians who laid the groundwork, from Shanks and Ke Schmid to the broader tapestry of number theory. This is number theory as a web of hidden connections—easy to state, deep to understand.

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