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.

Project Euler: I began the project by influence of James McMahon 17 Feb 2025. PM me if you want to friend there.

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.

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.)

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About

Married father of 2 adult children, St. Louis City, MO resident, native of Joliet, IL, Roman Catholic, avid competitive open water swimmer, fond of sketching, workshop, coffee. Fluent in Italian; know Spanish, French, Russian, and some Arabic. Crown scholar 1988 and Scarlet Hawk (alumnus of Illinois Institute of Technology), professional bachelor of architecture 1993. Self employed since 2004.