## 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 = prodki = 1pi\alphapi. The authors call n strongly prime-additive if n = sumkk = 1pi\betai, pi\betai < n \leq pi\betai+1. We only know three strongly prime additive numbers 228, 3115, 190233. n is prime additive if n = sumki = 1pi\gammai, 0 < \gammai \leq \betai. 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\deltarir, 0 < \deltar where pi1,... 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 log3x < A(x) < x/\exp (log x) ½-\epsilon.

A. Balog and C. Pomerance proved (2) A(x) > logkx 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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