OEIS A000377: Theta and Eta — The Multiplicative Mystery of A000377

Discover A000377, a seemingly simple integer sequence whose average converges to π/√6. This episode unpacks how the sequence is born from Ramanujan theta functions and Dedekind eta quotients, and how these two complex definitions yield the same integers. We explain its multiplicativity (a(2n)=a(n), a(3n)=a(n)), the prime-power rules depending on p mod 24 (e+1 when p ≡ 1,5,7,11 mod 24; alternating 0/1 when p ≡ 13,17,19,23 mod 24), and why zeros appear. We’ll connect to modular forms, the modular-24 arithmetic that governs the prime behavior, and the famous 42 in Martin’s table (1996) that anchors its eta-quotient identity. We’ll also sketch alternative definitions: the Möbius transform of a period-24 sequence and the Euler transform of another period-24 sequence, all yielding the same a(n). All of this culminates in the remarkable fact that the asymptotic mean of the sequence is π/√6, a vivid example of deep number-theoretic structure hiding in simple integers.

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