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**Zentralblatt MATH**

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

**Zbl.No: ** 695.10040

**Autor: ** Erdös, Paul; Ivic, Aleksandar

**Title: ** On the iterates of the enumerating function of finite Abelian groups. (In English)

**Source: ** Bull., Cl. Sci. Math. Nat., Sci. Math. 17, 13-22 (1989).

**Review: ** Let a(n) denote the number of non-isomorphic Abelian groups of order n. It was proved by the reviewer [Q. J. Math., Oxf. II. Ser. 21, 273-275 (1970; Zbl 206.03402)] that **limsup**_{n ––> oo}\frac{log a(n) log log n}{log n} = \frac{log 5}{4}. Now the authors investigate the iterates of a(n), which are defined by a^{(r)}(n) = a(a^{(r-1)}(n)), a^{(1)}(n) = a(n), r = 2,3,... . The main result is

a^{(2)}(n) << \exp**{**B(log n)^{7/8}/(log log n)^{19/16}**}** with a positive constant B and log a^{(r)}(n) << (log n)^{cr} with c_{1} = 1, c_{2} = 7/8 and c_{r} \leq (½)c_{r-1}+(3/8)c_{r-2} for r \geq 3.

Furthermore, let K(n) = **max** **{**r: a^{(r)}(n) = 1**}**. Then an asymptotic representation for the mean value of K(n) is established.

**Reviewer: ** E.Krätzel

**Classif.: ** * 11N45 Asymptotic results on counting functions for other structures

**Keywords: ** arithmetic functions; finite abelian groups; number of non-isomorphic Abelian groups of order n; iterates; asymptotic representation; mean value

**Citations: ** Zbl 206.034

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