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For    k  ⩾  2   , we consider the numberA  k    (  Z  )of positive integers    n  ⩽  Z    such that both   n   and    n  +  1    are   k  ‐free. We prove an asymptotic formulaA  k    (  Z  )   =   c  k   Z  +  O   (   Z   14  /  9  k  +  ϵ    )    , where the error term improves upon previously known estimates. The main tool used is the approximative determinant method of Heath‐Brown.

The density of twins of k ‐free numbers