Four dimensional N=1 supersymmetrization of R⁴ in superspace

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We write an action, in four dimensional N=1 curved superspace, which contains a pure R⁴ term with a coupling constant. Starting from the off-shell solution of the Bianchi identities, we compute the on-shell torsions and curvatures with this term. We show that their complete solution includes, for some of them, an infinite series in the R⁴ coupling constant, which can only be computed iteratively. We explicitly compute the superspace torsions and curvatures up to second order in this coupling constant. Finally, we comment on the lifting of this result to higher dimensions.