OEIS A000134: Zeros of the Bessel function J0 rounded to the nearest integer

Today we explore A000134, the sequence formed by rounding the positive zeros of the Bessel function J0 to the nearest integer. The actual zeros are approximately 2.4048, 5.5201, 8.6537, 11.7915, 14.9309, …, so the rounded values are 2, 6, 9, 12, 15, … The terms grow roughly linearly with pi, reflecting the asymptotic spacing z_n ~ (n − 1/4)π and a_n ≈ round(z_n). There is no known simple closed form for a_n. The OEIS page notes an interesting question: are the differences a_{n+1} − a_n always 3 or 4? It’s a small question with a surprising connection to how zeros of Bessel functions align on the number line. Beyond the numbers, these zeros underpin many applications—from vibrating circular membranes to cylindrical waveguides—and the rounded sequence provides a compact bridge between analysis and arithmetic. Finally, we’ll touch on how these zeros are computed with high precision and why understanding their distribution matters in modeling physical systems.

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