Certain numbers on the groups of self-homotopy equivalences

For a connected based space X  , let [X,X][X,X] be the set of all based homotopy classes of base point preserving self map of X   and let E(X)E(X) be the group of self-homotopy equivalences of X  . We denote by A♯k(X) the set of homotopy classes of self-maps of X   that induce an automorphism of πi(X)πi(X) for i=0,1,⋯,ki=0,1,⋯,k. That is, [f]∈A♯k(X) if and only if πi(f):πi(X)→πi(X)πi(f):πi(X)→πi(X) is an isomorphism for i=0,1,⋯,ki=0,1,⋯,k. Then, E(X)⊆A♯k(X)⊆[X,X] for a nonnegative integer k. Moreover, for a connected CW-complex X  , we have E(X)=A♯(X)E(X)=A♯(X). In this paper, we study the properties of A♯k(X) and discuss the conditions under which E(X)=A♯k(X) and the minimum value of such k. Furthermore, we determine the value of k for various spaces, including spheres, products of spaces, and Moore spaces.