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On prime-additive numbers. (English) Zbl 0688.10043

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 \(A(x)\).

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
11A41 Primes
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