practical numbers

[Richard Mathar]

> From seqfan-owner at ext.jussieu.fr  Mon Nov 27 17:07:28 2006
> Date: Mon, 27 Nov 2006 08:05:13 -0800
> From: "Tanya Khovanova" <tanyakh at tanyakhovanova.com>
> Reply-To: <tanyakh at tanyakhovanova.com>
> To: <seqfan at ext.jussieu.fr>
> Subject: practical numbers
> ...
> There is some confusion with practical numbers. Wikipedia article on practical numbers: http://en.wikipedia.org/wiki/Practical_numbe
> r
> gives the following definition of a practical number:
> A practical number or panarithmic number is a positive integer n such that all preceding positive integers are a sum of distinct div
> isors of n.
> 
> At the same time it refers to the sequence A005153 which gives a slightly different definition:
> Practical numbers (first definition): all k <= sigma(n) are sums of distinct divisors of n. Also called panarithmic numbers.
> 
> Browsing other websites (including my own :-) ) I've found that the wiki definition is used more often. Moreover, my calculation wit
> h the my definition exactly matches the sequence A005153, though they are supposed to be different.
> 
> Please, clarify my confusion.
> ...

The A005153 definition [with reference to the sigma(n)] is also used
in
http://www.dm.unipi.it/gauss-pages/melfi/public_html/articoli/jnt.ps
but the same author uses in his web page
http://members.unine.ch/giuseppe.melfi/pratica.html
just the other definition.
see also http://citeseer.ist.psu.edu/285.html
http://arxiv.org/abs/math.NT/0404555

B. M Stewart in Am J. Math 76 (4) (1954) p 779, available from
http://www.jstor.org, quotes Srinivasan [Current Science (1948), pp 179] with
the definition without the sigma.

E J Scourfield in J. Number Theory 62 (1) (1997) p 163 uses the definition
including the sigma:
http://www.emis.de/projects/EULER/detail?ide=1997saiaentidividens&matchno=7&matchtotal=18&q=cr%3AE*+Saias+

I have to stop here. The recent IAU resolution on what a useful definition of a
"planet" might be, in particular the status of Pluto, be has already caused
enough confusion, and we're still recovering from this.

Richard