User:Michael De Vlieger

I specialize in Wolfram code (Mathematica 14+), visualization, bfiles, and enter citations for OEIS Foundation.

In service of OEIS I sometimes author sequences mentioned in papers. If you are an author of such a paper, are registered here at OEIS, and would like attribution, I am happy to consider turning over authorship of such sequences to you, pending approval by other Editors in Chief. Simply contact me using this wiki. It is better for other researchers interested in your sequence to contact you rather than me, it is better aligned with where the sequence ideas originated.

General: Registered architect, interested in the relationship of multisets of prime factors of natural numbers. Joined OEIS in June 2014. Mathematica user since 2007. Contributed over 10000 Mathematica programs to OEIS and enjoy helping program sequences.

Some Visualizations

I've produced Mathematica visuals (sometimes mildly edited by photo-editor) of certain sequences. These are recent samples.

My favorite plots:

• Plot union of A2182 and A4394, $a(n)$ at $(x,y)=(a(n)/A2110(\omega (a(n)),\omega (a(n)))$, with a color function related to membership in sequences, particularly A166981, see legend at lower right.
• Diagram montage of rotationally symmetrical XOR triangles A334769.
• Graph of squarefree $m$ at $(x,y)=(\pi (gpf(m))),\phi (m)/m)$.
• Animation of prime power factors of “Idaho” numbers A347284.

Scatterplots with color functions that represent prime power decomposition of a(n) or n. These are usually log log scatterplots. The color functions may aid in forming hypotheses about the sequences, e.g., the pattern of emergence of primes or powerful numbers. Examples:

• A379248.
• A374284.
• A372699.
• A365000.
• A372368.

These use the following color functions:

Simple prime power decomposition legend (Legend 0):

Standard prime power decomposition legend (Legend 1):

There are also more advanced and finer legends that are spelled out in the sequences that employ them.

Fan style trees: These illustrate potential patterns that may be inherited from previous terms, usually from parent $a(n/2-n\mod 2)$:

• Binary: Doudna/A5940, A187769, A309840, A329697 (using color legend 1), A336384, A356321, A370470, A378299, A379762.
• Ternary: A356867.
• Heat map: A120945.

Fan style trees developed in 2022 from genealogy research, attempting to automate pedigree diagrams, and can be made to show any number of children and a variety of functions.

Large bitmaps that can compactify millions of terms of sequences that are composed of 0's and 1's. Example: A115517, A178788. Code can be supplied that can read these images to yield millions of terms of the sequence. Similar treatment for decimal digits A361340, A361338.

Color maps that represent the prime power decomposition ("constitution") of a number, similar to the binary bitmaps. Using legend 1 above: A372699, A357910.

Plot >p | a(n) at (x, y) = (n, &pi(p)). These plots study patterns among prime divisors of a sequence as n increases. A371572, 32X vertical exaggeration, A372007, 12X vertical exaggeration. Similar plot, A030723 a(w(j) + k - 1) at (j,k) for j = 1..512 and w the sequence of partial sums of A030719, showing a(m) = 1 in red and a(m) > 1 in light blue..

Plot p^m | a(n) at (x, y) = (n, &pi(p)), with a color function representing m, where m = 1 is black, m = 2 is red, m = 3 is orange, ..., largest m in the dataset in magenta. Similar to above, this style of plot demonstrates patterns among prime power factors of a(n) as n increases. A370974, 4X vertical exaggeration, A370968, 4X vertical exaggeration, A362855, 12X vertical exaggeration. At times these graphs employ a sort of “index” that employs a barcode like band below the main graph using legend 0 or 1 above. A textual version of this graph appears in the example at A372368.

Hasse diagrams: A376847, A334184.

Labeled plots:

• Color coded list employing legend 3 (also highlighting highly composite numbers, products of primorials, and even squarefree semiprimes). Converted into a color map: A367683.
• Diagram showing numbers k in A362010 instead as k mod 42, labeled and in large black circles, else gray dots if coprime to 42, purple if k = 1, red if k | 42, and gold if rad(k) | 42
• Plot $a(n)$ at $(x,y)=(a(n)/A2110(\omega (a(n)),\omega (a(n)))$, with $n$ labeled, and a color function showing a(n) ∈ A2182 in gold, a(n) ∈ A2201 in orange, and a(n) ∉ A2182 in red (see legend).
• A379753, similar color function to immediately above, see legend.

Color coded diagrams that relate the multisets of prime factors of two natural numbers k and n, using legend 9. A378984, A378984, A379336, A378900.

“Constitutive” relationship symbols and colors (Legend 9):

Special diagrams:

• Chart depicting prime power decomposition of A324581 vs. A002182.
• Graph of squarefree $m$ at $(x,y)=(\pi (gpf(m))),\phi (m)/m)$.
• Chart showing recursively self-conjugate partitions corresponding to $n$ in A323034.
• Expanded chart immediately above.
• Plot of 1 ≤ k ≤ 1200 in rows 1 ≤ n ≤ 34 of A322457, also relating A190900.

Original Sequences

My interests include sequences having to do with the multiset of prime factors of natural numbers, their relationships (especially {m : rad(m) | n}, sometimes called generalized n-regular numbers.)

Search

Research

See ResearchGate, orcID 0009-0002-0111-1194.

My interest regards elementary number theory, particularly the nature of the multiset of prime factors of natural numbers and their relations. The thesis includes Constitutive Basics with follow up investigations Constitutive State Counting Functions, Regular and Coregular Numbers, [Constitutive Quadrisection of Tantus (A126706). This research attributes to attempting to create a method of multiplication in base 60 without using the multiplication table (2007) and realizing relationships described in Hardy & Wright relating to expansion of fractions in various bases. This strain of interest had its start in 1989 (age 19) and is the reason for the original attraction to the Encyclopedia of Integer Sequences then (for the highly composite numbers and Euler's totient function).

Current research topics:

• "Constitutive" relations between codivisor (complementary divisor) pairs (d, n/d), which include "unitary divisors" and "coreful divisors". Have reference domains like A375055 and A376936, cardinality functions across reference domains like A379552 and A379752, and records transforms like A379553 and A379753. These have an interesting but understandable coincidence with A002182, as does A059992. This paper is expected to be complete in February 2025, underway since November 2024, and will appear at ResearchGate.

• Comparison of row n of A162306 with row n of A376248. Subject of August-October 2024. Interrupted by the immediately above.

• “Minimum theta”, minima of A010846 given a number k ∈ A126706 with ω(k) = m (θ = A010846 after Granville). See A376248, A376567, A376846, A376847, A377070, A372209, A374873. Subject of June-August 2024, interrupted by the immediately above.

• In 2025 I plan to post papers at arXiv, beginning with an old paper on A279818 edited, reformatted, and amplified to include variants like A378360.

Number Bases

I've been fascinated with number bases since fifth grade. Because I've been afflicted with this fascination for 40+ years. 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.

“Argam numerals” I invented between 1983 and 2008 (first 60 numerals - PDF: [1], first 360 - JPG: [2]).

I wrote an article for ACM Inroads in 2012: "Exploring number bases as tools" [3].

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

Machine Specs and Programming Languages

Current machine: “VinciExha” (2019.12.10) Dell Precision 7740. Intel(R) Xeon(R) E-2286M 64 bit CPU @ 2.40GHz, 64 Gb RAM, Windows 10 Pro for Workstations, NVIDIA Quadro RTX 4000. I sometimes use CUDA, but the graphics card is for visual and cinematic work.

I programmed BASIC since age 10 (between 1980-1994), FORTRAN between 1988 and 1993, HTML and Javascript, etc. between 1998 and 2006. Started programming Mathematica/Wolfram language in January 2007.

My Work

My company website is vincico.com[4] 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, 2011-2020. Crown scholar 1988 and alumnus of Illinois Institute of Technology, professional bachelor of architecture 1993. Self employed since 2004.