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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 688.10043

**Autor: ** Erdös, Pál; Hegyvári, Norbert

**Title: ** On prime-additive numbers. (In English)

**Source: ** Stud. Sci. Math. Hung. 27, No.1-2, 207-212 (1992).

**Review: ** Let n = **prod**^{k}_{i = 1}p_{i}^{\alphapi}. The authors call n strongly prime-additive if n = **sum**^{k}_{k = 1}p_{i}^{\betai}, p_{i}^{\betai} < n \leq p_{i}^{\betai+1}. We only know three strongly prime additive numbers 228, 3115, 190233. n is prime additive if n = **sum**^{k}_{i = 1}p_{i}^{\gammai}, 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^{\deltar}_{ir}, 0 < \delta_{r} where p_{i1},... 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 (1) c log^{3}x < A(x) < x/\exp (log x)^{ ½-\epsilon}. A. Balog and C. Pomerance proved (2) A(x) > log^{k}x for every k. It might be of some interest to get better inequalities for A(x).

**Reviewer: ** P.Erdös

**Classif.: ** * 11P32 Additive questions involving primes

11A41 Elemementary prime number theory

**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

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