Paul A. Loomis, Ph.D.

Associate Professor of Mathematics
Bloomsburg University

Ben Franklin Hall 204
(570) 389-4621

Classes for Fall 2014:

MATH 101 Trigonometry
MATH 126 Calculus 2
MATH 226 Calculus 4
HONORS 110 Math and Science I

Other courses I teach regularly:

MATH 113 Pre-Calculus
MATH 125 Calculus 1
MATH 185 Discrete Mathematics
MATH 225 Calculus 3
MATH 310 Abstract Algebra
MATH 360 Number Theory
MATH 421 Advanced Calculus (Real Analysis)

Other courses I have taught, but not that recently:

MATH 111 Finite Mathematics
MATH 109 College Algebra
MATH 123 Essentials of Calculus
MATH 241 Probability and Statistics
MATH 314 Linear Algebra
MATH 322 Differential Equations


New Families of Solitary Numbers,
to appear in Journal of Algebra and its Applications

Two Results on Sums of Cubes,
The Mathematical Gazette 95 (2011), 506-510.

(with Florian Luca) On Totient Abundant Numbers,
INTEGERS: Electronic Journal of Combinatorial Nmumber Theory 8 (2008), Article A6, 1-7.

(with Michael Plytage and John Polhill) Summing Up the Euler Phi Function,
College Mathematics Journal 39 (2008), 34-42.

An Introduction to Digit Product Sequences,
Journal of Recreational Mathematics 32 (2003-2004), 147-151.

Degree Two Generalized Iteration of q-Additive Polynomials,
in Algebra, Arithmetic and Geometry with Applications (2004), 601-608.

(with Matthew Severcool) The Density of Abundant Numbers,
Proceedings of SSHEMA, 2003.

(with Mehdi Razzaghi) The Concept of Hormesis in Developmental Toxicology,
Human and Ecological Risk Assessment 7 (2001), 933-942.

(with S.S. Abhyankar) Equations of Similitude,
Proceedings of the Indian Academy of Sciences 109 (1999), 1-9.

(with S.S. Abhyankar) Twice More Nice Equations for Nice Groups,
Contemporary Mathematics 245 (1999), 63-76.

(with S.S. Abhyankar) Once More Nice Equations for Nice Groups,
Proceedings of the American Mathematical Society 126 (1998), 1885-1896.

Other creative stuff:

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 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

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

(these won't make much sense unless you've read the article):

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).

Mathematical Links:

My old vita

Bloomsburg University

the Department of Mathematics, Computer Science, and Statistics

e-math, the home page of the American Mathematical Society

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 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.

Frivolous Links:

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.