The saturation number of a graph F, written $$\text{ sat }(n,F)$$ sat ( n , F ) , is the minimum number of edges in an n-vertex F-saturated graph. One of the earliest results on saturation numbers is due to Erdős et al. who determined $$\text{ sat }(n,K_r)$$ sat ( n , K r ) for all $$r \ge 3$$ r ≥ 3 . Since then, saturation numbers of various graphs and hypergraphs have been studied. Motivated by Alon and Shikhelman’s generalized Turán function, Kritschgau et al. defined $$\text{ sat }(n,H,F)$$ sat ( n , H , F ) to be the minimum number of copies of H in an n-vertex F-saturated graph. They proved, among other things, that $$\text{ sat }(n,C_3,C_{2k}) = 0$$ sat ( n , C 3 , C 2 k ) = 0 for all $$k \ge 3$$ k ≥ 3 and $$n \ge 2k +2$$ n ≥ 2 k + 2 . We extend this result to all odd cycles by proving that for any odd integer $$r \ge 5$$ r ≥ 5 , $$\text{ sat }(n , C_r ,C_{2k} ) = 0$$ sat ( n , C r , C 2 k ) = 0 for all $$2k \ge r+5$$ 2 k ≥ r + 5 and $$n \ge 2kr$$ n ≥ 2 k r .