Unique wavenumbers are calculated for axisymmetric Rayleigh–Be´nard convection as a function of the Rayleigh number (R) up to the second critical value for several different Prandtl numbers. The analysis assumes slightly bent rolls (large radius of curvature) and that there exists a horizontal pressure gradient strong enough to force the net mean flow induced by curvature to be zero. The assumptions are satisfied for axisymmetric convection in a large aspect ratio cylinder (however, this may not be the only case). Manneville and Piquemal [Phys. Rev. A28, 1774 (1983)] found the initial slope of the selected wavenumber with respect to Rayleigh number (using an analytic solution valid for small amplitude solutions) and our calculations agree with theirs. This initial slope is sensitive to the Prandtl number (P), but at moderate to largeRthe selected wavenumber is approximately independent ofPwhenP>3. For smallerPlarger wavenumbers are found, but this does not contradict any available experimental evidence.