I am on sabbatical in Argentina from June 2007 to May 2008. You can still email me at the address above.

A few notes on Summing Up the Euler Phi Function, in the January 2008 issue of the College Mathematics Journal

First, the sum-of-totients function (labelled capital Phi in our paper and F in the following) can be implemented much more efficiently in Mathematica as follows:

F[x_]:=Total[FixedPointList[EulerPhi,x]]-x-1;

The first possible divergent sequence F^i[n] occurs when n=107; we know that the 10,981st term of this sequence is about 1.4 x 10^45, and that this is not exceeded until the 266,819th term. The largest known element in the sequence is the 267,024th term, which is 14,881,235,108,935,957,540,595,506,098,516,550,746,038,890,495. This is not exceeded in the first 350,000 terms.

Since our final revisions, we have become aware of two more papers related to perfect totient numbers (PTNs), both from the Journal of Integer Sequences, 2006.

Igor Shparlinski, On the Sum of the Iterations of the Euler Function, Vol. 9, Article 06.1.6. Most pertinent to our paper, Shparlinski proves that the PTNs have density 0.

Florian Luca, On the Distribution of Perfect Totients, Vol. 9, Article 06.4.4. Among other results, Luca proves that the totient abundant numbers (those for which F(n)>n) also have density 0. Also, the sum of the reciprocals of all PTNs converges.

Mathematical Links:

My old vita

Bloomsburg University

the Department of Mathematics, Computer Science, and Statistics

MAA online, for the Mathematical Association of America

Project NExT

The math genealogy project or who's your grandfather?

The Erdos Number Project Home Page. If you've published a paper with a mathematician, chances are you have a finite Erdos number. (Mine is 3.)

The math department at Purdue, my alma mater

An old page on some iterated sequences.

Neal Sloane's Online Encyclopedia of Integer Sequences.. Sequences AO63108, AO63112, AO63113, AO63114, and AO63425 all pertain to the iterated sequences mentioned above.

Frivolous Links:

I should mention that Dry Ridge, my first album of original folk songs, was released in 2005. You can get a copy at Muggsy's in Bloomsburg or Schoolhouse Music in Danville. Or, you can go to CDBaby to buy or hear snippets of songs. The songs are also on Napster, iTunes, and about 20 more similar sites.

Webpages are sometimes like a messy back porch - if you look around, you'll find all kinds of things lying there, some half finished, some discarded years ago but never thrown away, because there's enough space. In that spirit, here is 1) a link to my running page, with several links that were alive 6 years ago and the vital statistics of every race I ran between 1989 and 2001, and 2) a few stories about vehicles that I have been lucky enough to own.