International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 2, Pages 373-383
doi:10.1155/S0161171293000456

Arithmetic functions associated with infinitary divisors of an integer

Graeme L. Cohen1,2 and Peter Hagis Jr.1,2

1School of Mathematical Sciences, University of Technology, Broadway, Sydney 2007, NSW, Australia
2Department of Mathematics, Temple University, Philadelphia 19122, PA, USA

Received 21 November 1991

Copyright © 1993 Graeme L. Cohen and Peter Hagis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The infinitary divisors of a natural number n are the products of its divisors of the form pyα2α, where py is a prime-power component of n and ∑αyα2α (where yα=0 or 1) is the binary representation of y. In this paper, we investigate the infinitary analogues of such familiar number theoretic functions as the divisor sum function, Euler's phi function and the Möbius function.