We study the positive subharmonic solutions to the second order nonlinear ordinary differential equationwhere g(u) has superlinear growth both at zero and at infinity, and q(t) is a T -periodic sign-changing weight. Under the sharp mean value condition T 0 q(t) dt < 0, combining Mawhin's coincidence degree theory with the Poincaré-Birkhoff fixed point theorem, we prove that there exist positive subharmonic solutions of order k for any large integer k. Moreover, when the negative part of q(t) is sufficiently large, using a topological approach still based on coincidence degree theory, we obtain the existence of positive subharmonics of order k for any integer k ≥ 2. * Work partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Progetto di Ricerca 2016: "Problemi differenziali non lineari: esistenza, molteplicità e proprietà qualitative delle soluzioni". It is also partially supported by the project "Existence and asymptotic behavior of solutions to systems of semilinear elliptic partial differential equations" (T.1110.14) of the Fonds de la Recherche Fondamentale Collective, Belgium.