To solve this problem, we need to analyze the state changes of each student based on their ID and the number of times their state is flipped.

Key Observations:

Each student's state (standing/sitting) is flipped once for every divr of their ID.

• Divrs of non-perfect squares: Come in pairs (e.g., 6 has divrs 1&6, 2&3 → even count of divrs).
• Divrs of perfect squares: Include a repeated divr (e.g., 4 has divrs 1&4, 2→ odd count of divrs).

Initial state: Standing.

• Even number of flips → returns to standing (original state).
• Odd number of flips → changes to sitting.

Calculation:

• Number of perfect squares from 1 to 1000: $\lfloor \sqrt{1000} \rfloor = 31$ (since $31^2=961$, $32^2=1024>1000$).
• Number of non-perfect squares (standing students): $1000 - 31 = 969$.

Answer: 969.

$\boxed{969}$

作者声明:本文包含人工智能生成内容。