I have made three albums of original folk songs: Dry Ridge (2005), Yuspe (2009), and World Famous in Bloomsburg (2014). There is stuff about this, like how to find me playing live or how to buy an album, at paulloomis.com. If you are feeling nostalgic, you can hear 5 entire (older) songs at my MySpace page . The songs are also on Napster, iTunes, and about 20 more similar sites.
In 2011 I wrote a song called Susquehanna (Here it comes again), which won 3rd prize in the Pennsylvania Heritage Songwriting Contest. I am playing (or, I played, if you are reading this after March 2013) this song as well as two songs by Van Wagner in Bloomsburg Theatre Ensemble's production of Flood Stories, Too.
I can most often be found playing music at Brews n Bytes in Danville, Center Street Cafe in Bloomsburg, or, when I can make the drive, the Open Mike at Godfrey Daniels in Bethlehem.
Along with my wife Leticia Weber and our kids Anna and Zeke, I spend six months of 2012 travelling in South America. Our blog of that trip - and occasional posts from regular life - can be found at weberloomis.wordpress.com.
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:
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.
Make that three more papers:
Paul Loomis and Florian Luca, On Totient Abundant Numbers, in Integers: Electronic Journal of Combinatorial Number Theory, Vol. 8 (2008), Artice A06. We show that the density of totient abundant numbers is 0, though this proportion goes to 0 very very very very slowly (that is, like 1/log log log log x).
My old vita
the Department of Mathematics, Computer Science, and Statisticse-math, the home page of the American Mathematical Society
MAA online, for the Mathematical Association of America
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 2.)
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.
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 2010, and 2) a few stories about vehicles that I have been lucky enough to own.