  We prove that the equation $(x-3r)^3+(x-2r)^3 + (x-r)^3 + x^3 + (x+r)^3 +
(x+2r)^3+(x+3r)^3= y^p$ only has solutions which satisfy $xy=0$ for $1\leq
r\leq 10^6$ and $p\geq 5$ prime. This article complements the work on the
equations $(x-r)^3 + x^3 + (x+r)^3 = y^p$ and $(x-2r)^3 + (x-r)^3 + x^3 +
(x+r)^3 + (x+2r)^3= y^p$ . The methodology in this paper makes use of the
Primitive Divisor Theorem due to Bilu, Hanrot and Voutier for a complete
resolution of the Diophantine equation.
