Note on the sum of cubes of primes and an almost prime (Q1364579)

Let \(E(N)\) be the number of positive integers \(n\in\mathbb{N}\) \((n\equiv 0(2)\), \(n\not\equiv\pm1,\pm3(9)\), \(n\not\equiv\pm 1(7))\) which are not representable in the form \(n= P^3_3+ p^3_1+p^3_2+ p^3_3\) (\(p_1\), \(p_2\), \(p_3\) denote primes, \(P_r\) denotes positive integers with at most \(r\) prime factors). It is shown that \(E(N)\ll N(\log N)^{-A}\) for any fixed \(A>0\). The proof follows the method of \textit{J. Brüdern} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 28, No. 4, 461-476 (1995; Zbl 0839.11045)], where the same result was gained with \(P_4\) instead of \(P_3\). The improvement is achieved by a different use of the linear sieve.