OEIS - Contributions

 https://www.insightest.app/apps/math/oeis/jim.html

When I create a sequence for the On-Line Encyclopedia of Integer Sequences (OEIS), I typically post it in this blog. A colleague of mine, Vincenzo Manto, created the above webpage with all my contributions.

MAGNIFICA HUMANITAS - Encyclical Letter of Pope Leo XIV

Pope Leo XIV

(Image by: Edgar Beltrán, The Pillar Deed - Attribution-ShareAlike 4.0 International - Creative Commons )

On May 25, 2026, Pope Leo XIV issued his first encyclical letter, Magnifica Humanitas. This was near the 135th anniversary of Rerum Novarum issued by Pope Leo XIII during a time of rapid technological change.

Why is this in a math blog? As many know, Pope Leo XIV, studied mathematics at Villanova University before pursuing theological studies and earning a JCD from Pontifical University of St. Thomas Aquinas in Rome. Pope Leo's recent letter is to address the rapid changes in society driven by artificial intelligence and related technologies. With his degree in mathematics, the Pope is in a good position to speak to the church about this topic.

Within the first few pages, his mathematical background comes out in the analogy: "This concept can also be illustrated by the image of a multifaceted polyhedron, in which the one truth of the Gospel is reflected from different angles."

While the encyclical is primarily theological and pastoral, his math influence seems most evident in additional sections:

1. Chapter 3: Technology and Dominance – The Grandeur of Humanity in Light of the Promises of AI (especially paragraphs on AI itself)

This is the most directly relevant section. The Pope offers a careful, almost analytical breakdown of what AI is and is not. He describes AI as systems that “imitate certain functions of human intelligence” through data, models, and optimization — language that echoes mathematical concepts like algorithms, statistical models, and pattern recognition.

• He stresses transparency regarding algorithms, independent checks, accountability, and the need to understand how systems classify people and situations. This reflects a mathematician’s insistence on verifiable processes, error analysis, and avoiding “black box” opacity.

• Discussions of bias in algorithms, data as a shared resource, and how models embed values (what they measure, ignore, or optimize) show systems-thinking typical of someone trained in applied mathematics.

2. Sections on Governance, Subsidiarity, and Ethical Regulation of AI

The encyclical repeatedly calls for transparency, accountability, independent verification, and structured participation in AI governance. These mirror mathematical and scientific habits: demanding clear assumptions, reproducible results, and checks against unintended consequences.

His emphasis on subsidiarity (handling issues at the most appropriate level) and avoiding top-down imposition of opaque systems feels informed by logical structuring of complex problems.

3. Discussions of Truth, Probability, and Decision-Making

In parts addressing truth as a common good, misinformation, and automated decision-making (e.g., credit, hiring, or risk assessment), the Pope highlights how algorithms can cloak exclusion in “a veneer of neutrality and objectivity.” This critique shows awareness of how mathematical tools can appear impartial while carrying hidden biases in their design or training data.

4. Broader Structural Approach

The encyclical’s overall organization — clear chapters, logical progression from foundations to applications, and balanced weighing of risks vs. benefits — reflects disciplined, systematic thinking. Some observers note that his math training may contribute to a more rigorous, less purely rhetorical style in addressing technical topics.

Notable Quote Reflecting Precision

One standout line (around paragraph 128) contrasts human growth with machine logic:

“For an algorithm, an error is a flaw to be corrected; for a person, however, an error can be a catalyst for profound change.”

This beautifully distinguishes deterministic systems (math/AI) from the open-ended, relational nature of human freedom and grace.

Overall Assessment: Pope Leo XIV does not engage in deep technical mathematics in the encyclical. Instead, his background seems to provide intellectual tools for dissecting AI as a complex system, insisting on clarity, ethical guardrails, and human-centered design. It helps him bridge theology and technology without being either overly fearful or naively optimistic.

His formation allows a precise critique: AI is powerful modeling, but it lacks the irreducible dignity, freedom, and relational depth of the human person created in God’s image.

Contribution to the OEIS - A395743

 A395743

Sum of the cumulative number of previous occurrences of the digits of n in the sequence 1..n excluding the current occurrence of each digit.

0

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 5, 6, 7, 8, 9, 10, 11, 12, 3, 15, 9, 8, 9, 10, 11, 12, 13, 14, 5, 17, 18, 13, 11, 12, 13, 14, 15, 16, 7, 19, 20, 21, 17, 14, 15, 16, 17, 18, 9, 21, 22, 23, 24, 21, 17, 18, 19, 20, 11, 23, 24, 25, 26, 27, 25, 20, 21, 22, 13

(list; graph; refs; listen; history; edit; text; internal format)

OFFSET

1,11

COMMENTS

This sequence is the exclusive counterpart to A343644. While A343644 counts the occurrences of digits in the range [1,n], a(n) counts only the occurrences strictly preceding each digit of n. Formally a(n) = A343644(n) - A055642(n). Thus, this sequence represents the exclusive scan of digit occurrences, whereas A343644 is the inclusive scan. a(n) = 0 for all single-digit n as it measures the cumulative repetition of digits at the exact moment n is formed.

LINKS

Vincenzo Manto, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = Sum_{i=1..k} C(d_i), where d_1, d_2, ..., d_k are the digits of n, and C(d_i) is the number of times digit d_i has appeared in the concatenation of all integers from 1 to n-1 plus the digits of n to the left of d_i.

a(n) = A343644(n) - A055642(n).

EXAMPLE

For n = 11:

The digits are '1' and '1'.

- First '1': Appeared previously in {1, 10} -> count = 2.

- Second '1': Appeared previously in {1, 10} AND as the first digit of 11 -> count = 3.

a(11) = 2 + 3 = 5.

For n = 12:

The digits are '1' and '2'.

- Digit '1': Appeared in {1, 10, 11 (twice)} -> count = 4.

- Digit '2': Appeared in {2} -> count = 1.

a(12) = 4 + 1 = 5.

MATHEMATICA

a[n_]:=Module[{sm=0, id=IntegerDigits[n], d=IntegerDigits/@Range[n-1]//Flatten}, Do[sm=sm+Count[Join[d, Take[id, i-1]], id[[i]]], {i, IntegerLength[n]}]; sm]; Array[a, 70]

PROG

(Python)

def generate_sequence(limit):

seq = []

counts = [0] * 10

for n in range(1, limit + 1):

v = 0

s = str(n)

for char in s:

d = ord(char) - 48

v += counts[d]

counts[d] += 1

seq.append(v)

return seq

print(generate_sequence(70))

CROSSREFS

Cf. A007953, A343644, A055642.

Sequence in context: A152481 A157276 A010486 * A298185 A329082 A111423

Adjacent sequences: A395738 A395740 A395741 * A395744 A395745 A395746

KEYWORD

nonn,base,easy,new

AUTHOR

Vincenzo Manto and James C. McMahon, May 05 2026

STATUS

approved

16670, A Saintly Number

 

(Image: By Unknown author - http://santuarioeucaristico.blogspot.hu/2010/08/sao-maximiliano-kolbe-14-de-agosto.html, Public Domain, https://commons.wikimedia.org/w/index.php?curid=42823235)

In 1941, a prisoner at the Auschwitz concentration camp escaped. In retaliation, the officials at the camp selected ten prisoners who would be starved to death. One of the selected prisoners, Franciszek Gajowniczek, a Polish Catholic, cried out, "My wife! My children!" Upon hearing his pleas, a Franciscan Friar, Maximilian Maria Kolbe, volunteered to take the place of Gajowniczek. 

In the following weeks, each time the guards checked on him, he was standing or kneeling in the middle of the cell, calmly looking at those who entered. After the group had been starved and deprived of water for two weeks, only Kolbe and three others remained alive.

Impatient to empty the bunker, the guards gave the four remaining prisoners lethal injections of carbolic acid. Kolbe is said to have raised his left arm and calmly waited for it. Maximilian Kolbe died on 14 August 1941. He was cremated on 15 August, which happened to be the feast day of the Assumption of Mary.

Kolbe was canonized by Pope John Paul II on 10 October 1982.

Gajowniczek was transferred from Auschwitz to Sachsenhausen concentration camp on 25 October 1944. He was liberated there by the Allies, after spending five years, five months, and nine days in concentration camps in total. He reunited with his wife Helena, six months later in Rawa Mazowiecka. 

Saint Maximilian Kolbe's prisoner number at Auschwitz was 16670.

Top Italian Mathematicians

 

(Image: Designed by Kubanek / Freepikw.freepik.com )

1. Fibonacci (Leonardo of Pisa, c. 1170–1240)
   Often regarded as the most influential medieval European mathematician. He introduced the Hindu-Arabic numeral system (including zero) to the Western world in his book Liber Abaci, revolutionizing commerce, science, and calculation. He is best known for the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, ...), which appears in nature, art, and modern applications like computer algorithms and biology.
   Also see: Fibonacci Day, Speed Limits, The Creator
   
   
   (Image: By Hans-Peter Postel - Own work, CC BY 2.5, https://commons.wikimedia.org/w/index.php?curid=1739679)
   
   
   
   
   
2. Joseph-Louis Lagrange (Giuseppe Luigi Lagrangia, 1736–1813)
   A towering figure in mathematical analysis, celestial mechanics, and number theory. Born in Turin (then part of the Kingdom of Sardinia), he worked extensively in France and is considered one of the greatest mathematicians of the 18th century. Key contributions include Lagrangian mechanics (foundational to classical physics), the Lagrange multiplier method in optimization, and major advances in algebra and the calculus of variations. He is frequently ranked among the top Italians in historical popularity indices.
   
   
   (Image: Public Domain - https://commons.wikimedia.org/wiki/File:Lagrange_crop.jpg)
   
   
3. Gerolamo Cardano (1501–1576)
   A Renaissance polymath whose Ars Magna (1545) was the first major Latin treatise on algebra, introducing solutions to cubic and quartic equations (building on earlier Italian work by del Ferro, Tartaglia, and Ferrari). He also made pioneering contributions to probability theory and is noted for his broad influence on mathematics, medicine, and philosophy during the Italian Renaissance.
   
   
   (Image: Ginko Edizioni)
   
   
   
4. Galileo Galilei (1564–1642)
   While primarily remembered as a physicist and astronomer, Galileo was a profound mathematician who applied rigorous mathematical methods to the study of motion, falling bodies, and kinematics—laying groundwork for modern physics and the scientific method. His work on geometry, proportions, and experimental mathematics bridged the Renaissance and the Scientific Revolution. Many lists of great Italian mathematicians include him for his mathematical innovations in mechanics.
   
   
   (Image: Public Domain - https://en.wikipedia.org/wiki/Galileo_Galilei#/media/File:Galileo_Galilei_(1564-1642)_RMG_BHC2700.tiff)
   
   
   
5. Vito Volterra (1860–1940) 
   Vito Volterra is renowned for his work in functional analysis (integral equations, Volterra operators) and mathematical biology (predator-prey models, now foundational in ecology).
   
   
   
   
   (Image: By Unknown author - http://www.phys.uniroma1.it/DipWeb/dottorato/SCUO_VOLTERRA/scuola_volterra.html, Public Domain, https://commons.wikimedia.org/w/index.php?curid=16117839 )

Other Notable Mentions

• Luca Pacioli (c. 1447–1517): "Father of accounting" and popularizer of double-entry bookkeeping; collaborator with Leonardo da Vinci.
• Bonaventura Cavalieri and Evangelista Torricelli: 17th-century pioneers in indivisibles and early calculus ideas.
• Scipione del Ferro, Niccolò Tartaglia, and Lodovico Ferrari: Solved cubic and quartic equations in the Renaissance.
• Maria Gaetana Agnesi (1718–1799): Early female mathematician known for the "witch of Agnesi" curve and comprehensive calculus text.
• Gregorio Ricci-Curbastro developed the tensor calculus (with Tullio Levi-Civita), which became essential for Einstein's general relativity and differential geometry. Other strong contenders for this spot include Giuseppe Peano (axiomatization of natural numbers and mathematical logic) and Ennio de Giorgi or Enrico Bombieri (Fields Medalist) for more modern contributions.

Modern-era figures like Enrico Bombieri (Fields Medal 1974) and Eugenio Calabi also rank highly in specialized fields but are less "all-time" dominant than the historical giants. Other current figures include:

• Alfio Quarteroni (born May 30, 1952, in Ripalta Cremasca, Italy) is a prominent Italian mathematician specializing in numerical analysis, scientific computing, and mathematical modeling.
• Professor Piergiorgio Odifreddi (born 1950 in Cuneo, Italy) is an Italian mathematician, logician, and popular science writer specializing in mathematical logic, recursion theory (computability theory), and the foundations of mathematics. He is Professor Emeritus of Mathematical Logic at the University of Turin, where he taught for many years, and has held visiting positions at institutions such as Cornell University and the University of California, Berkeley.
• Professor Roberto Renò (born in Italy) is an Italian mathematician and quantitative finance expert specializing in financial econometrics, volatility modeling, asset pricing, and statistical methods for financial markets. He holds a PhD in Financial Mathematics from the Scuola Normale Superiore in Pisa (2005, magna cum laude) and a degree in Physics from the University of Pisa.

100,000!

This blog recently reached 100,000 views. The number 100,000 is significant in many ways.

• In South Asia, one hundred thousand is one lakh and is expressed numerically as 1,00,000.
• Cead Mile Failte is an Irish greeting meaning 100,000 welcomes.
• 100,000 meters is the altitude considered where space flight begins.
• 104,723 and 104,729 are the 9999th and 10,000th prime numbers (by the prime number theorem, the frequency of prime numbers decreases as numbers increase - it makes it easy to remember that the frequency is 1/10 at 100,000).
• In the Netherlands, 100,000 is informally called a ton.
• More interesting facts about 100,000 at: https://en.wikipedia.org/wiki/100,000 

A392975 - A contribution to the On-Line Encyclopedia of Integer Sequences (OEIS)

 

(Image: Michael De Vlieger)

I recently proposed a new sequence for the On-Line Encyclopedia of Integer Sequences (OEIS). 

A Sisyphus sequence: a(0) = 0, a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive integer having the same greatest prime factor as the sum of the previous terms.

Sisyphus is a figure in Greek mythology, known as the cunning King of Corinth who was punished by Zeus to eternally roll a massive boulder up a hill in Tartarus, only for it to roll back down each time. Numerical sequences are sometimes called Sisyphus sequences when their terms climb and climb, but then drop repeatedly. 

The image above shows this process for this sequence for the first 10^6 terms. Below is a graph for how the sequence begins (first 300 terms):

One interesting feature of the sequence is the low points all appear to be prime. They begin:  7, 13, 17, 19, 23, 29, 59, 107, 137, 173, 257, 293, 467, 503...

This trend continues for at least the first 200,000 terms of the sequence.

Another feature of the sequence is that beginning with the third term, the common greatest prime factor repeats in runs of 3 or more. These factors begin:  3, 3, 3, 7, 7, 7, 7, 13, 13, 13, 13... and this trend continues for at least the first 200,000 terms of the sequence.

(Image: Sisyphus (1548–49) by Titian, Prado Museum, Madrid, Spain)

Update 4/3/2026: the sequence was published A392975 - OEIS .

Update 4/14/2026: a companion sequence, A396326, was published: Low points in A392975 (having the property of being all prime numbers).

Volumes in Higher Dimensions

(Image: Freepik)

A prior post of this blog, Atomic Interstitial Sizes in Higher Dimensions, explored how the size of gaps between tightly packed spheres changes as one moves from two-dimensional to three-dimensional and on to higher dimensions.

Grant Sanderson, host of the 3Blue1Brown YouTube channel (I highly recommend following this channel), gave an exceptional lecture explaining how to calculate the volume of high-dimensional spheres. That lecture is: https://www.youtube.com/watch?v=fsLh-NYhOoU&t=3467s. While he doesn't directly speak about the interstitial sizes of the gaps found in my post, he does approach the problem by looking at the ratio of the sphere to cube volume in higher dimensions.