Some numbers are missing from S!

the number of the form $4k+2$ have been removed from $S$ . There are exactly $24$ numbers for each type $4k$ , $4k+1$ and $4k+3$ . If the number $i$ is of either $4k$ or $4k+3$ types, then $\pi \sum_{j=1}^{i} j$ is an even number and, consequently its cosine is $1$ . If the number $i$ is of type $4k+1$ , then $\pi \sum_{j=1}^{i} j$ is an odd number and, consequently its cosine is $-1$ . There are $24 \times 2$ ones and $24$ minus ones, so, they add up to $24$ .