Formality of Eₙ-algebras and cochains on spheres

We study the loop and suspension functors on the category of augmented Eₙ-algebras. One application is to the formality of the cochain algebra of the n-sphere. We show that it is formal as an Eₙ-algebra, also with coefficients in general commutative ring spectra, but rarely E₍+₁-formal unless the coefficients are rational. Along the way we show that the free functor from operads in spectra to monads in spectra is fully faithful on a nice subcategory of operads which in particular contains the stable Eₙ-operads for finite n. We use this to interpret our results on loop and suspension functors of augmented algebras in operadic terms.