Different steps leading to the new functional for pairing based on natural orbitals and occupancies proposed in ref. [D. Lacroix and G. Hupin, arXiv:1003.2860] are carefully analyzed. Properties of quasi-particle states projected onto good particle number are first reviewed. These properties are used (i) to prove the existence of such a functional (ii) to provide an explicit functional through a 1/N expansion starting from the BCS approach (iii) to give a compact form of the functional summing up all orders in the expansion. The functional is benchmarked in the case of the picked fence pairing Hamiltonian where even and odd systems, using blocking technique are studied, at various particle number and coupling strength, with uniform and random single-particle level spacing. In all cases, a very good agreement is found with a deviation inferior to 1% compared to the exact energy.