Erdős, Pál; Hegyvári, Norbert On prime-additive numbers. (English) Zbl 0688.10043 Stud. Sci. Math. Hung. 27, No. 1-2, 207-212 (1992). Let \(n=\prod^{k}_{i=1}p_i^{\alpha_{p_i}}\). The authors call \(n\) strongly prime-additive if \(n=\sum^{k}_{k=1}p_i^{\beta_i}\), \(p_i^{\beta_i}<n\leq p_i^{\beta_i+1}\). We only know three strongly prime additive numbers 228, 3115, 190233. \(n\) is prime additive if \(n=\sum^{k}_{i=1}p_i^{\gamma_i}\), \(0<\gamma_i\leq \beta_i\). We do not know if there are infinitely many prime additive numbers. \(n\) is weakly prime additive if it is not power of a prime, and \(n=\sum p^{\delta_r}_{i_r}\), \(0<\delta_r\) where \(p_{i_1},\ldots\) is a subset of the prime factor of \(n\). We prove that there are infinitely many weakly prime-additive numbers. Denote by \(A(x)\) the number of the weakly prime-additive numbers not exceeding \(x\). We prove \[ c \log^3x<A(x)<x/\exp (\log x)^{1/2-\varepsilon}. \tag{1} \] A. Balog and C. Pomerance proved (2) \(A(x)>\log^kx\) for every \(k\). It might be of some interest to get better inequalities for \(A(x)\). Reviewer: Pál Erdős (Budapest) Cited in 1 ReviewCited in 3 Documents MSC: 11P32 Goldbach-type theorems; other additive questions involving primes 11A41 Primes Keywords:counting functions; strongly prime additive numbers; weakly prime-additive numbers; de Bruijn’s function; Hardy-Ramanujan theorem; representation of a number by powers of its prime divisors PDFBibTeX XMLCite \textit{P. Erdős} and \textit{N. Hegyvári}, Stud. Sci. Math. Hung. 27, No. 1--2, 207--212 (1992; Zbl 0688.10043) Online Encyclopedia of Integer Sequences: Weakly prime-additive numbers: numbers n with at least 2 distinct prime factors that can be represented as n = Sum_{some p|n} p^e_p with e_p > 0. Prime-additive numbers: numbers n with at least 2 distinct prime factors, that can be represented as n = Sum_{p|n} p^e_p, with e_p > 0.