Given a positive integer k, find the least integer nk for which there exist five sets S1,S2,S3,S4,S5 with the following properties:
|S_j|=k \text{ for } j=1, \cdots , 5 , |\bigcup_{j=1}^{5} S_j | = n_k ;
|S_i \cap S_{i+1}| = 0 = |S_5 \cap S_1|, \text{for } i=1,\cdots ,4 floor functionceiling functionnumber theory unsolvednumber theory