Summary: A Hilbert space operator \(A\in \mathcal{B}(H)\) is a generalised \(n\)-projection, denoted \(A\in (G-n-P)\), if \({A^*}^n=A\). \((G-n-P)\)-operators \(A\) are normal operators with finitely countable spectra \(\sigma(A)\), subsets of the set \(\{0\}\cup\{\sqrt[n+1]{1}\} \). The Aluthge transform \(\tilde{A}\) of \(A\in \mathcal{B}(H)\) may be \((G-n-P)\) without \(A\) being \((G-n-P)\). For doubly commuting operators \(A, B\in \mathcal{B}(H)\) such that \(\sigma(AB)=\sigma(A)\sigma(B)\) and \(\|A\|\|B\|\leq \left\|\widetilde{AB}\right\|\), \(\widetilde{AB}\in (G-n-P)\) if and only if \(A=\left\|\tilde{A}\right\|(A_{00}\oplus(A_0\oplus A_u))\) and \(B=\left\|\tilde{B}\right\|(B_0\oplus B_u)\), where \(A_{00}\) and \(B_0\), and \(A_0\oplus A_u\) and \(B_u\), doubly commute, \(A_{00}B_0\) and \(A_0\) are 2 nilpotent, \(A_u\) and \(B_u\) are unitaries, \(A^{*n}_u=A_u\) and \(B^{*n}_u=B_u\). Furthermore, a necessary and sufficient condition for the operators \(\alpha A, \beta B, \alpha \tilde{A}\) and \(\beta \tilde{B}, \alpha=\frac{1}{\left\|\tilde{A}\right\|}\) and \(\beta=\frac{1}{\left\|\tilde{B}\right\|} \), to be \((G-n-P)\) is that \(A\) and \(B\) are spectrally normaloid at \(0\).