On the r-free values of the polynomial x2 + y2 + z2 + k

Let k be a fixed integer. We study the asymptotic formula of R ( H,r,k ), which is the number of positive integer solutions 1 ⩽ x, y, z ⩽ H such that the polynomial x 2 + y 2 + z 2 + k is r -free. We obtained the asymptotic formula of R ( H, r, k ) for all r ⩾ 2. Our result is new even in the case r = 2. We proved that R ( H , 2, k ) = c k H 3 + O ( H 9/4+ ε ), where c k > 0 is a constant depending on k . This improves upon the error term O ( H 7/3+ ε ) obtained by G.-L. Zhou, Y. Ding (2022).