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Divisibilityhard
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Let v2(x)v_2(x)v2​(x) denote the exponent of the highest power of 2 dividing xxx. Evaluate the sum S=∑i=12024v2(i2+2i)S = \sum_{i=1}^{2024} v_2(i^2 + 2i)S=∑i=12024​v2​(i2+2i).