User:Michael De Vlieger

I specialize in Wolfram code (Mathematica 13+), visualization, bfiles, and run citations for NJAS.

In service of OEIS I sometimes enter 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 and users of OEIS to contact you rather than me, and it is better aligned with where the sequence ideas originated.

Registered architect and lover of integer number bases. Joined OEIS in June 2014. Mathematica user since 2008. Contributed about 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.

• Plot of terms $m$ in A325237 at $(x,y)=(\pi (gpf(m))),\phi (m)/m)$.
• Plot of terms $m$ in A325236 at $(x,y)=(\pi (gpf(m))),\phi (m)/m)$.
• Chart depicting prime power decomposition of A324581 vs. A002182.
• Compare A059894 (blue) with A307544 (red); coincident points (green).
• Graph of squarefree $m$ at $(x,y)=(\pi (gpf(m))),\phi (m)/m)$.
• Compare A307113 (blue) with A307327 (red).
• Graph.
• Graph of A306237$n$ 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 terms of A323034 (black) in A321223(n) (color) for $n\leq 65536$.
• Chart depicting recursively self-conjugate partitions with side length $n$ for numbers $m$ in A322457.
• Plot of $1\leq k\leq 1200inrows<math>1\leq n\leq 34$ of A322457, also relating A190900.
• Chart showing recursively self-conjugate partitions of $m$ based on Durfee square of side length $n$.
• Chart of recursively self-conjugate partitions for $1\leq n\leq 150$.
• Color-coded graph of A321223 showing relation $n(\mod 3).$.
• Graph of highly composite and superabundant numbers.
• Graph of 779,674 HCNs $m$ according to $p\#,m/p\#$ where $p\#$ is the largest primorial dividing $m$.

Original Sequences (120)

Sequences I've written relate to regular numbers $k|n^{\epsilon }with\epsilon \geq 0$, the Euler totient function, nondivisors in the cototient of $n$, numbers neither coprime to $n$ nor regular, recursively self-conjugate partitions, highly composite and superabundant numbers, and methods of encoding prime decomposition.

• A325237: Squarefree $k$ such that ${\frac {1}{2}}-\phi (k)/k$ is positive and minimal for $k$ with $gpf(k)=prime(n)$.
• A325236: Squarefree $k$ such that $\phi (k)/k-{\frac {1}{2}}$ is positive and minimal for $k$ with $gpf(k)=prime(n)$.
• A307544: Binary encoding of A307540: $T(n,k)=A087207(A307540(n,k))$.
• A307540: Irregular triangle $T(n,k)$ such that squarefree $m$ with $gpf(m)=prime(n)$ in each row are arranged according to increasing values of $\phi (k)/k$.
• A306237: Primorial A002110(n)/A002110(n - 1).
• A307327: Number of superabundant numbers ($m$ in A004394) in the interval $p_{k}\#\leq m<p_{(k+1)}\#$, where $p_{i}\#=A002110(i)$. (Analogous to A307113).
• A307322: Irregular triangle where row $n$ is a list of indices in A002110 with multiplicity whose product is $A004394(n)$. (Analogous to A306737.)
• A307107: $a(n)=A025487(n)/A247451(n)$.
• A307133: $T(n,m)=$ number of $k\leq A002110(n$ such that $A001222(k)=m$, where $k$ is a term in A025487.
• A307113: Number of highly composite numbers ($m$ in A002182) in the interval $p_{k}\#\leq m<p_{(k+1)}\#$, where $p_{i}\#=A002110(i)$.
• A307056: Row $n$ = digits of $A025487(n)$ in primorial base.
• A306802: Position of highly composite numbers in the sequence of products of primorials. (Analogous to A293635.)
• A306737: Irregular triangle where row $n$ is a list of indices in A002110 with multiplicity whose product is $A002182(n)$.
• A323035: Records in A321223.
• A323034: Where records occur in A321223.
• A322457: Irregular triangle: Row $n$ contains numbers $k$ that have recursively symmetrical partitions having Durfee square with side length $n$.
• A322156: Irregular triangle where row $n$ includes all decreasing sequences $S=\{k_{0}=n,k_{1},k_{2},\ldots ,k_{m}\}$ in reverse lexical order such that the sum of subsequent terms $k_{j}$ for all $i<j\leq m$ does not exceed any $k_{i}$.
• A321223: $a(n)$ is the number of recursively self-conjugate partitions of $n$.
• A305056: $a(n)=A004394(n)/A002110(A001221(A004394(n)))$. (For superabundant $m$, $a(n)=m/p_{\omega (m)}\#$).
• A305025: $a(n)=A001221(A004394(n))=\omega (m)$ where $m$ is superabundant.
• A304886: Irregular triangle where row $n$ contains indices $k$ where the product of $A002110(k)=A025487(n)$.
• A304235: Colossally abundant numbers that are highly composite, but not superior highly composite.
• A304234: Superior highly composite numbers that are superabundant but not colossally abundant.
• A301415: Numbers $k$ in A301413 such that $k\times A002110(m)$ is in A002201.
• A301414: Numbers $k$ in A301413 such that $k\times A002110(m)$ is in A002182.
• A301413: $a(n)=A002182(n)/A002110(A108602(n))$. (For highly composite $m$, $a(n)=m/p_{\omega (m)}\#$).
• A301893 Numbers $m$ that set records for the ratio $A010846(m)/A000005(m)$. This is the ratio of the regular counting function and the divisor counting function, where $k|n^{e}withe\geq 0$, integers, is regular to $n$ and counted when $k\leq n$.
• A301892 $a(n)=A010846(A002182(n))$. Number of regular $k|m^{e}withe\geq 0,k\leq m$ for highly composite $m$.
• A300914: Records in A045763.
• A300861: Records in A300858.
• A300860: Indices of records in A300858.
• A300859: Where records occur in A045763. Highly Neutral Numbers.
• A300858: a(n) = A243823(n) − A243822(n).
• A300157: Records in A299990.
• A300156: Indices of records in A299990.
• A300155: Numbers $n$ for which $A243822(n)=A000005(n)$ (i.e., $\xi _{d}(n)=\tau (n)$).
• A299992: Composite $n$ with $\omega (n)>1$ for which $A243822(n)<A000005(n)$.
• A299991: Numbers $n$ for which $A243822(n)>A000005(n)$ (i.e., $\xi _{d}(n)>\tau (n)$).
• A299990: a(n) = A243822(n) − A000005(n) = $\xi _{d}(n)-\tau (n)$.
• A294168: Irregular triangle read by rows in which row $n$ contains significant digits after the radix point for unit fractions ${\frac {1}{n}}$ expanded in factorial base.
• A295523: Nonprime numbers $n$ such that $A243822(n)\geq A243823(n)$, i.e., $\xi _{d}(n)\geq \xi _{t}(n)$.
• A295221: Numbers $k$ such that $2\times A243823(k)=k$.
• A294576: Odd $m$ such that $2\times A243823(m)>m$.
• A294575: Numbers $m$ such that $2\times A243823(m)>m$.
• A294492: Numbers $n$ that set records for the ratio $A045763(n)/n$. (i.e., the ratio $\xi (n)/n$).
• A294306: Irregular triangle $T(n,m)$ = total number of each value $k$ in row $n$ of A280269.
• A293556: Records in A243822.
• A293555: Indices of records in A243822.
• A292868: Records in A243823.
• A292867: Indices of records in A243823.
• A292393: Base-$n$ digit $k$ involved in anomalous cancellation in the proper fraction $A292288(n)/A292289(n)$.
• A292289: Smallest denominator of a proper fraction that has a nontrivial anomalous cancellation in base $b$.
• A292288: Numerators of smallest denominator of a proper fraction that has a nontrivial anomalous cancellation in base $b$.
• A291928: Positions of records in A218320.
• A291927: Records transform of A218320.
• A291834: Positions of records of A252665.
• A291833: Records transform of A252665.
• A291213: Start from the singleton set $S=\{n\}$, and unless 1 is already a member of $S$, generate on each iteration a new set where each odd number $k$ is replaced by $3k+1$, and each even number $k$ is replaced by $3k+1$ and ${\frac {k}{2}}$. $a(n)$ is the total size of the set from the singleton through after the first iteration which has produced 1 as a member, inclusive.
• A289172: Irregular triangle read by rows: row $n$ lists terms $m$ of $A038566(n)$ such that $\omega (m)=A051265(n)$, with $a(1)=1$.
• A289171: Irregular triangle $T(n,k)$ read by rows with $1\leq k\leq n$: $T(n,1)$ = A020900(n − k + 1) − (n − k + 1) and T(n, k) = max(0, T(n − 1, k − 1) − 1) otherwise.
• A288813: Irregular triangle read by rows: $T(m,k)$ is the list of squarefree numbers A002110(m) < t < 2 A002110(m + 1) such that $\omega (t)=m$.
• A288784: Irregular triangle read by rows: $T(n,m)$ is the list of numbers k × A002110(n) ≤ k × t < (k + 1)A002110(n) such that $\omega (k\times t)=n$, with $1\leq k<prime(n+1)$.
• A287692: Triangle read by rows: $T(n,k)$ is the greatest difference between prime factors among squarefree numbers A002110(n) ≤ m ≤ (A002110(n + 1) − 1) such that $\omega (m)=n$ and $m$ is divisible by A002110(k).
• A287484: Triangle $T(n,k)$: number of squarefree numbers A002110(n) ≤ m ≤ (A002110(n + 1) − 1) such that $\omega (m)=n$ and $m$ is divisible by A002110(k).
• A287483: Squarefree numbers A002110(n) ≤ m ≤ (A002110(n + 1) − 1) such that $\omega (k)=n$.
• A287352: Irregular triangle $T(n,k)=A112798(n,1)$ followed by first differences of $A112798(n)$.
• A287010: Triangle $T(n,m)$: $\lfloor (\log p_{n}\#)/(\log prime(m))\rfloor$.
• A286424: Number of partitions of $p_{n}\#$ into parts $(q,k)$ both coprime to $p_{n}\#$, with $q$ prime and $k$ nonprime, where $p_{n}\#=A002110(n)$.
• A286300: Square root of smallest square formed from $n$ by incorporating all the digits of $n$ in a new decimal number.
• A285905: $a(n)=A275768(A002110(n))$.
• A285904: Partial row products of table A027746, prime factors with repetition, reversed.
• A285788: Irregular triangle $T(n,m)$: nonprime $1\leq k\leq n$ such that $\gcd(n,k)=1$.
• A285769: (Product of distinct prime factors)^(Product of prime exponents).
• A284061: Triangle read by rows: $T(n,k)=\pi (p_{k}\times p_{(n+1)})$.
• A283866: Multiplicities of prime factors of A243103(n).
• A280363: $a(n)=\lfloor \log _{p}n\rfloor withe=A020639(n)$, the least prime factor of $n$.
• A280274: $a(n)=$ maximum value in row $n$ of A279907 (Also, maximum value in row $n$ of A280269).
• A280269: Irregular triangle $T(n,k)$ read by rows: least $\rho$ such that $r|n^{\rho }$ applied to terms $r$ in row $n$ of A162306.
• A279907: Triangle read by rows: $T(n,k)$ = smallest power of $n$ that is divisible by $k$, or $-1$ if no such power exists.
• A277071: Numbers $n$ for which $A277070(n)$ does not equal $A237442(n)$.
• A277070: Row length of $A276380(n)$.
• A277045: Irregular triangle $T(n,k)$ read by rows giving the number of partitions of length $k$ such that all of the members of the partition are distinct and in A003586.
• A276380: Irregular triangle where row $n$ contains terms $k$ of the partition of $n$ produced by greedy algorithm such that all elements are in A003586.
• A276379: Write a 1 for each distinct prime divisor $p|n$ in the (primepi(p) − 1)-th place, ignoring multiplicity.
• A275881: Numbers $n$ such that $A010846(n)\geq n/2$.
• A275280: Irregular triangle listing numbers $m\leq n$ such that $m|n^{\epsilon }$ with $\epsilon \geq 0$, in order of appearance in a matrix of products that arranges the power range of each $p$ along independent axes. (Algorithm of A010846, akin to A275055).
• A275055: Irregular triangle read by rows listing divisors $d|n$ in order of appearance in a matrix of products that arranges the power ranges of each $p|n$ along independent axes.
• A273258: Write the distinct prime divisors $p|n$ in place (primepi(p) − 1), ignoring multiplicity. Decode the resulting number after first reversing the code, ignoring any leading zeros.
• A272619: Irregular array read by rows: $n$-th row contains (in ascending order) the numbers $1\leq k<n$ such that at least one prime divisor $p|k$ also divides $n$ and at least one prime divisor $q|k$ is coprime to $n$.
• A272618: Irregular array read by rows: $n$-th row contains (in ascending order) numbers $k<n$ such that $k|n^{\epsilon }$ with $\epsilon >1$.
• A262115: Digits of the base-$b$ expansion of ${\frac {1}{7}}$.
• A262114: Digits of the base-$b$ expansion of ${\frac {1}{5}}$.
• A256577: Raise decimal digit k to the $\epsilon$ power, where $10^{\epsilon }$ is the place value.
• A254336: Powers of 10 written in base 60, concatenating the decimal values of sexagesimal digits.
• A254335: Powers of 5 written in base 60, concatenating the decimal values of sexagesimal digits.
• A254334: Powers of 3 written in base 60, concatenating the decimal values of sexagesimal digits.
• A250089: 5-smooth numbers written in base 60, concatenating the decimal values of sexagesimal digits.
• A250073: Powers of 2 written in base 60, concatenating the decimal values of sexagesimal digits.
• A243103: Product of $n$-regular numbers $m\leq n$ such that $m|n^{\epsilon }$ with $\epsilon \geq 0$.
• A242028: Numbers $k$ such that the least common multiple of the anti-divisors of $k$ is less than $k$.
• A241557: Numbers $k$ that do not have prime anti-divisors.
• A241556: Number of prime anti-divisors $m$ of $n$.
• A241419: Number of numbers $m\leq n$ that have one prime divisor $p>{\sqrt {n}}$ such that $p|m$.
• A245500: Concatenation of multiplicities of prime divisors of highly composite numbers $A002182(n)$.
• A244974: (Regular Sum Function) Sum of numbers $m\leq n$ such that $m|n^{\epsilon }$ for $\epsilon \geq 0$.
• A244053: Records in A010846.
• A244052: Recordsetters in A010846 (Highly Regular Numbers).
• A243823: Number of $1\leq k<n$ such that at least one prime divisor $p|k$ also divides $n$ and at least one prime divisor $q|k$ is coprime to $n$. (Semitotative Counting Function $\xi _{t}(n)$).
• A243822: Number of $k\leq n$ such that $k|n^{\epsilon }$ for $\epsilon >1$. (Semidivisor Counting Function $\xi _{d}(n)$).

Handy Sequences

• A067255 Exponents e of $p_{n}^{e}$ written in the n-th place, abbreviated MN(n).
• A287352 "π-code" (pi-code), first differences of exponents of the prime divisors p arranged in order of magnitude of p from least to greatest. Abbreviated PC(n).
• A027746 The prime factors of n with multiplicity.
• A027748 The distinct prime factors of n.
• A001221 (little) ω(n) = number of distinct prime factors of n
• A001222 (big) Ω(n) = number of distinct prime factors of n counting multiplicity.
• A020639 Least prime divisor of n.
• A006530 Greatest prime divisor of n.
• A002110 the primorials, products $p_{n}\#$ of the smallest n primes.
• A060735 $k\times p_{n}\#$ with $1\leq k\leq p_{(n+1)}$.
• A007947 squarefree root of n: largest squarefree number k | n.
• A025487 products of primorials (least integer of each prime signature).
• A124010 prime signature of n.
• A280363 floor(log_p n}, with p the least prime that divides n.

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.

Check out my multiplication tables for each base between 2 and 60 inclusive (8 Mb PDF) [1]. 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: [2], first 360 - JPG: [3]).

I wrote an article for ACM Inroads in 2012: "Exploring number bases as tools" [4].

Map of the number-theoretical qualities of numbers m <= n of bases n: [5]. 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 2018.

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.

Planned machine in late 2023: Dell Precision 7760. “VinciRyna”: Intel Core i7-11850H @ 2.50-4.80GHz, 64 Gb RAM, Windows 11, with NVIDIA RTX A3000.

I have 5 machines in the company, VinciExha has Mathematica, Adobe products, and CAD/BIM software for other purposes. I have used Dell since 1998 and Precision since 2007. Before this I had a Macintosh PowerBook 180c (the first portable color Mac) in 1994, an Apple Macintosh (original) in 1984, and an Apple II+ in 1980 (at age 10).

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

In February 2022 I began learning C++. I am looking for faster programming capabilities.

My Work

My company website is vincico.com[6] 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.