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# User:Michael De Vlieger

### From OeisWiki

Registered architect and lover of integer number bases. Joined OEIS in June 2014. Mathematica user since 2008. Contributed over 4300 Mathematica programs to OEIS and enjoy helping program sequences.

## Contents |

## Mathematica

Since January 2015 I've modeled my Wolfram language coding at the OEIS on that of Robert G. Wilson v and others and endeavor to write succinct and efficient code for your sequences. I quite enjoy writing code for sequences I understand, but will usually defer to anyone else who's written Mathematica code for a sequence.

The code I've written between July 2014 and January 2015 typically generates the OEIS sequence up to a limit n via a function a###. These were intended to be self contained functions that are labeled such that a user may cut and paste the function without worrying that it will collide with another function they'd already defined in their notebook. This also has the advantage of noting the OEIS sequence that supplied the code. I have abandoned the practice in February 2015; when these codes are up for editing I will replace them with more straightforward code. Thereafter I used the latest version to write code up until mid spring 2016. Generally stopped using the latest functions because it isn't as accessible to people as are slightly older constructs. I tend to use anonymous functions, maybe too much. Feel free to tell me what you think.

I will add a little guide to the Version 10 functions I most often use soon. These functions are {AllTrue, AnyTrue, Nothing, SelectFirst, FirstPosition}. Many of these can be rendered using pre-10 command structures like Times @@ Boole@ f == 1, Total@ Boole@ f > 0, etc. I have gone back to this for the most part even though AllTrue, etc., may be more succinct in language.

I've written some graphically magnificent Manipulate functions regarding number bases and may post them to Wolfram Demonstrations when I get the nerve. Wrote these before I began contributing here.

## Sequences

Check out sequences I contributed here: [1] (via OEIS search) and background on them at my website [2].

Topics:
Concatenation of multiplicities of prime divisors of highly composite numbers A245500.

Positional notation akin to A058482 denoting prime divisors of n: A276379, notation reversed and decoded: A273258.

Multiple-base number system:
Greedy algorithm producing a partition of n such that all elements are unique and in A003586: A276380, row sums A277070. Numbers n such that A277070(n) != A237442(n): A277071. Irregular triangle T(n,k) of the number of partitions of length k such that all of the members of the partition are unique and in A003586 (numbers of all possible DBNS representations of n of length k): A277045.

Tensor products of power ranges of prime divisors of n bounded by n: power ranges 1..p^e | n: A275055, power ranges p^e <= n: A275280. Divisors A275055.

Regular numbers A243103, A244052, A244053, A244974 (cf. A010846, A162302), A275280, 275881, smallest power of n that regular k divides ("richness" of k with respect to n) A279907, multiplicities of factors of A243103: A283866.
Semidivisors A243822, A272618; semitotatives A243823, A272619. (two types of neutral number, cf. A045763, A133995).

Antidivisors (A241556, A241557, A242028, A242029).

Powers of some numbers in base 60: A250073, A254334, A254335, A254336.

Hamming numbers in base 60: A250089.

Decimal-base arithmetic A256577.

Full Reptend Primes: A261773.

Digits of base-b representation of 1/5 A262114, 1/7 A262115.

Collaboration:
With Robert G. Wilson v:
Smallest n-digit prime having at least n-1 digits equal to x: A268701-A268709.

Number of n-digit primes in which n-1 of the digits are x's. A266141-A266149.

With Jean-Marc Falcoz: A257350.

My favorite sequences have to do with "regular" and "neutral" numbers, A244052, and many of Elemer Labos' sequences. (An integer m regular to integer n if all the prime divisors p of m also divide n, and an integer a is neutral to b if 1 < GCD(a, b) < a.) I have a lot of sequences that I would like to contribute but rarely contribute as many are perhaps not of general interest. I am influenced by user Robert Munafo's thoughts on contributing sequences. I recommend reading his thoughts on sequences, as well as those of others.

Check out some proofs I wrote about neutral numbers (cf. A045763) here: [3]. A "semidivisor" is a number m < n whose prime divisors p also divide n, but m does not divide n. A "semitotative" is a number m < n that has at least one prime factor p that divides n and at least one prime factor q that is coprime to n. Both are composite, cototient to n, and nondivisors of n. Divisors and semidivisors, considered together, constitute the regular numbers m < n of n, as seen in A

Take a look at proofs I wrote about antidivisors here: [4].

## Number Bases

I've been fascinated with number bases since fifth grade. Because I've been afflicted with this fascination for 36 years, I know a lot about them. It has made me an "odd duck" among creatives, but also very efficient in my early career, conducting field measurements in US Customary measure rendered "metric" by using mixed radix arithmetic for onsite verification. This fascination is mellowing into a sober love of number theory.

Check out my multiplication tables for each base between 2 and 60 inclusive (8 Mb PDF) [5]. Recently these tables were generated with Mathematica so they are absolutely correct. (Please tell me if they are not and I will make them right). They use argam numerals for "transdecimal" bases, i.e., those bases b > 10. An adaptation of these charts is downloaded hundreds of times a month on another website.

"Argam numerals" I invented between 1983 and 2008 (first 60 numerals - PDF: [6], first 360 - JPG: [7]).

I wrote an article for *ACM Inroads* in 2012: "Exploring number bases as tools" [8].

Map of the number-theoretical qualities of numbers m <= n of bases n: [9]. I am making this map into a poster. It visually illustrates numbers m that divide n, that are coprime to n, that are regular but do not divide n, etc.

My favorite bases are 12, 60, and 120 and am pretty fluent in them in that order.

I am writing Mathematica code for an automatically generated website that regards integer number bases. This is the "Number Base" project; I've written the number theory engine and register. It should be complete in mid 2017.

## My Work

My company website is vincico.com[10] with examples of my "day job" work: modeling construction worksites. Since it's all digitally modeled, the work is sometimes surprisingly intertwined in number theory (spacings, divisions of spans, etc.) The work typically comes together with little upfront information, do-or-die deadlines 2-3 weeks ahead of notification, and rapidly evolving directives. The goal is usually to win a construction bid or to inform user groups of work on site. It often requires "filling in missing pieces" under urgent deadline pressure, and this is where I attempt to use highly divisible numbers that mesh with construction standard modules (often 4, 7, 12, 16, 48, 120, etc. inches) present and particular to the job to shorten development time. I am looking to further automate the digital modeling process of worksites - this is a move shared by several of us in the industry. One day I hope to use Wolfram language in the models.

## About

Married father of 2 children (daughter 2003 and son 2007), St. Louis City, MO resident, native of Joliet, IL, Roman Catholic, avid swimmer, fond of sketching, workshop, coffee. Fluent in Italian; know Spanish, French, Russian, and some Arabic. Overexcitable ENTP. FIRST Robotics mentor, algebra tutor. Crown scholar 1988 and alumnus of Illinois Institute of Technology, professional bachelor of architecture 1993. Self employed since 2004.