  We call indexed-biharmonic maps, the solutions of a particular non linear
elliptic PDE of order 4. This is a generalization of harmonic maps which
verifies that biharmonic maps are biharmonic of index 0. The goal of this
article is to study submanifolds of $\mathbb{S}^n$ whose inclusion is non
harmonic and indexed-biharmonic (i.e. proper indexed-biharmonic). We give some
pseudo-umbilical and non pseudo-umbilical examples and we study the link
between the proper indexed-biharmonic submanifolds of $\mathbb{S}^n$ and the
proper conformal-harmonic submanifolds of $\mathbb{S}^n$.
