For each set \(A\subset \{1,2,\ldots,n\}\), let \(A^*=\{\sum_{b\in B}b: B\subseteq A\}\). Let \(p(n,r)\), \(r\geq 2\) (resp. \(f(n,m)\), \(m\geq 1\)) be such that \(A^*\) contains no \(r\)th power of an integer (resp. such that \(m\not\in A^*)\). Estimates on asymptotic formulae for \(p(n,r)\) and \(f(n,m)\) are obtained; a special case gives a conjecture of Erdős and Graham that \(f(n,m)\sim (\tfrac 12+o(1))(n/\log n)\). The proofs rely on a technical result which is established using the Hardy-Littlewood method.