A385288 - Contribution to the OEIS: Numbers with a prime number of prime factors, counted with multiplicity, and whose prime factors are each raised to a prime exponent

 A385288

Numbers with a prime number of prime factors, counted with multiplicity, and whose prime factors are each raised to a prime exponent.

0

4, 8, 9, 25, 27, 32, 49, 72, 108, 121, 125, 128, 169, 200, 243, 288, 289, 343, 361, 392, 500, 529, 675, 800, 841, 961, 968, 972, 1125, 1323, 1331, 1352, 1369, 1372, 1568, 1681, 1800, 1849, 2048, 2187, 2197, 2209, 2312, 2700, 2809, 2888, 3087, 3125, 3267, 3481

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

OFFSET

1,1

COMMENTS

a(n) = A114129(n) through n=25; then a(26) = 961 and A114129(26) = 864.

Subset of A056166.

Subset of A001694. - Michael De Vlieger, Jun 25 2025.

LINKS

James C. McMahon, Table of n, a(n) for n = 1..1000

EXAMPLE

200 = 2^3 * 5^2; 200 has a prime number of prime factors, counted with multiplicity (3 + 2 = 5), and exponents 3 and 2 are prime.

MATHEMATICA

Select[Range[10^4], AllTrue[Last/@FactorInteger[#], PrimeQ]&&PrimeQ[PrimeOmega[#]]&]

PROG

(PARI) isok(k) = my(f=factor(k)); isprime(bigomega(k)) && (sum(k=1, #f~, isprime(f[k, 2])) == omega(f)); \\ Michel Marcus, Jun 25 2025

CROSSREFS

Cf. A001248, A001694, A030078, A056166, A114129.

Sequence in context: A383211 A051676 A114129 * A383103 A053810 A076702

Adjacent sequences: A385283 A385284 A385286 * A385289 A385292 A385293

KEYWORD

nonn,new

AUTHOR

James C. McMahon, Jun 24 2025

STATUS

approved

Online Encyclopedia of Integer Sequences, OEIS.

Multiple Dimensions of Time - Part 4

(Graphic: Carbon Design on Iconfinder)

Another article has been published about time existing in multiple dimensions. Gunther Kletetschka recently published Three-Dimensional Time: A Mathematical Framework for Fundamental Physics in Reports in Advances of Physical Sciences, Volume 09, 2025.

Prior posts on this subject include:

Multiple Dimensions of Time

Dimensions of time raised in science fiction: Robert Heinlein book, The Pursuit of the Pankera (The Pursuit of the Pankera | Arc Manor Books)

Multiple Dimensions of Time - Part 2:

Dynamical topological phase realized in a trapped-ion quantum simulator | Nature

Multiple Dimensions of Time - Part 3:

Relativity of superluminal observers in 1 + 3 spacetime

Vera C. Rubin Observatory

 

(Image: https://noirlab.edu/public/images/iotw2229a/)

The Vera C. Rubin Observatory, located at an altitude of 2700 m in Chile, recently became operational. The observatory is named for Vera C. Rubin, an American astronomer who pioneered discoveries about galactic rotation rates (this discovery has led to the understanding of dark matter).

Site construction began on 14 April 2015 with the ceremonial laying of the first stone. The first on-sky observations with the engineering camera occurred on 24 October 2024, while system first light images were released 23 June 2025. Images are recorded by a 3.2-gigapixel charge-coupled device imaging (CCD) camera, the largest digital camera ever constructed.

Some of its first released images are below:

(Photos: NSF–DOE Vera C. Rubin Observatory)

Here is a video of asteroids detected by the new telescope: https://www.youtube.com/watch?v=DTuq-vBsDJE&t=45s

X = 500 (Space X completes 500th mission)

 

(Graphic: https://www.iconfinder.com/bogdanrosu)

Later today (6/12/2025), if the launch goes as scheduled, Space X will have launched their 500th mission. The vast majority of their missions have been made with the Falcon 9 platform which employs the re-useable first-stage booster. Space X has drastically reduced the cost of launching payloads into space by reusing the first stage. One such booster, B1062, was used 25 times.

As of this date,  SpaceX accounted for approximately 73% of global space launches.

Space X Falcon 9

800,000,000,000 Stars

 

(Image: NASA, ESA, CSA, STScI )

In the image above, the James Webb Space Telescope (JWST) captures the Sombrero galaxy in near-infrared and mid-infrared wavelengths. This galaxy is estimated to have approximately 800,000,000,000 stars.

The image above compares the galaxy in infrared and visible wavelengths. 

Read more from the JWST imaging of the Sombrero galaxy at: https://www.nasa.gov/image-article/webb-sees-sombrero-galaxy-in-near-infrared/

Prior posts on JWST:

Number of planets in the universe:

https://jamesmacmath.blogspot.com/2022/01/number-of-planets-in-universe.html

Earliest star formation:

https://jamesmacmath.blogspot.com/2025/05/jwst-detects-earliest-formed-galaxies.html

Top ten JWST photographs:

https://jamesmacmath.blogspot.com/2024/07/top-ten-photographs-of-james-webb.html

Famous Mathematicians of Vietnam

 

Based on available information and their prominence in global mathematics, here are five notable Vietnamese mathematicians:

Ngô Bảo Châu (b. 1972)

Renowned for proving the fundamental lemma for automorphic forms, he became the first Vietnamese to win the Fields Medal in 2010, often regarded as the highest honor in mathematics. His work has had a significant impact on algebraic geometry, and he is a professor at the University of Chicago.

(Bởi Oberwolfach Photo Collection – https://opc.mfo.de/detail?photo_id=9920, CC BY-SA 2.0 de, https://commons.wikimedia.org/w/index.php?curid=11234718)

Hoàng Xuân Sính (b. 1933)

A pioneering figure, she was the first female mathematics professor in Vietnam and a student of the influential mathematician Alexander Grothendieck. Her work in mathematics education and research during challenging times, including the Vietnam War, has left a lasting legacy.

(https://golem.ph.utexas.edu/category/2022/06/hong_xun_snh.html)

Phạm Tuấn Huy

Recognized as the second Vietnamese mathematician to receive a Clay Research Fellowship, following Ngô Bảo Châu. His contributions to mathematics, though less detailed in public records, mark him as an emerging figure in the global mathematical community.

(https://stanfordsciencefellows.stanford.edu/people/huy-tuan-pham)

Phan Thanh Nam

Honored in 2020 as the first Vietnamese mathematician to receive a prestigious international award (specific details vary), his work in mathematical physics has gained attention, adding to Vietnam’s growing mathematical reputation.

(https://www.mcqst.de/about/members/phan-thanh-nam.html)

Lương Thế Vinh (1441–1496)

A historical scholar and mathematician from the Lê dynasty, he is celebrated in Vietnam for his contributions to mathematics education and problem-solving, including the compilation of mathematical texts that influenced Vietnamese scholarship.

(Bởi Không rõ – http://thltvq1.edu.vn/tin-tuc/index.php?option=com_content&view=article&id=326:tieu-su-luong-the-vinh&catid=103:truyen-thong&Itemid=467, Phạm vi công cộng, https://commons.wikimedia.org/w/index.php?curid=40677055)

Famous Brazilian Mathematicians

Brazil has produced many notable mathematicians who have made significant contributions to various fields of mathematics. Below is a list of ten prominent Brazilian mathematicians, based on their recognition, contributions, and impact, as derived from available information. Note that "fame" is subjective, and this list prioritizes those with international recognition, significant awards, or historical importance, while acknowledging that comprehensive data on all Brazilian mathematicians is limited.

1.  Artur Avila (b. 1979)  

   -Field: Dynamical systems, spectral theory, chaos theory.  

   -Notable Achievements: First Latin American to win the Fields Medal (2014), recognized for his work in dynamical systems. His contributions have brought clarity to the field of chaos theory, earning him global acclaim. 

(https://en.wikipedia.org/wiki/Artur_Avila)(https://americasquarterly.org/fulltextarticle/aq-top-5-latin-american-academics-artur-avila/)
(https://twas.org/article/brazilian-artur-avila-wins-twas-lenovo-prize)

   -Why Famous: The Fields Medal, often considered the "Nobel Prize of mathematics," marks him as Brazil’s most celebrated contemporary mathematician.

2. Jacob Palis (1940–2025)

   - Field: Dynamical systems.  

   - Notable Achievements: Internationally recognized for his work in dynamical systems, served as president of the International Mathematical Union, and received numerous awards, including the Balzan Prize.

(https://impa.br/en/notices/morre-matematico-jacob-palis-gigante-da-ciencia-brasileira/)
(https://www.famousfix.com/list/21st-century-brazilian-mathematicians)

   - Why Famous: A giant in Brazilian science, he held influential roles and mentored many mathematicians.

3. Júlio César de Mello e Souza (1895–1974)

   - Field: Mathematics education, recreational mathematics.  

   - Notable Achievements: Known by his pseudonym Malba Tahan, he wrote *The Man Who Counted*, a popular book blending mathematics with storytelling, widely influential in Brazil and beyond.

(https://proofwiki.org/wiki/Mathematician:Mathematicians/Sorted_By_Nation/Brazil)
(https://www.famousfix.com/list/brazilian-mathematicians)

   - Why Famous: His accessible works popularized mathematics in Brazil.

4. Manfredo Perdigão do Carmo (1928–2018)

   - Field: Differential geometry.  

   - Notable Achievements: Renowned for his work in differential geometry, particularly on Riemannian manifolds, and authored influential textbooks used worldwide.

 (https://proofwiki.org/wiki/Mathematician:Mathematicians/Sorted_By_Nation/Brazil)

   - Why Famous: His textbooks and research contributions have a lasting impact on global mathematics education.

5. Elon Lages Lima (1929–2017)

   - Field: Topology, mathematics education.  

   - Notable Achievements**: A leading figure in Brazilian mathematics, he contributed to topology and wrote numerous textbooks that shaped mathematics education in Brazil. 

 (https://proofwiki.org/wiki/Mathematician:Mathematicians/Sorted_By_Nation/Brazil)

   - Why Famous: His educational contributions made him a household name in Brazilian academia.

6. Paulo Ribenboim (b. 1928)

   - Field: Number theory.  

   - Notable Achievements: Known for his work in number theory, particularly on prime numbers and Fermat’s Last Theorem, and authored several influential books.  

(https://proofwiki.org/wiki/Mathematician:Mathematicians/Sorted_By_Nation/Brazil)

   - Why Famous: His extensive publications and international career (notably in Canada) brought him recognition.

7. Fernando Codá Marques (b. 1979)

   - Field: Differential geometry, geometric analysis.  

   - Notable Achievements: Recognized for solving major conjectures in geometric analysis, he received the ICTP Ramanujan Prize and other honors.

(https://www.famousfix.com/list/21st-century-brazilian-mathematicians)(https://www.famousfix.com/list/brazilian-mathematicians)

   - Why Famous: His groundbreaking work has earned him international acclaim among modern mathematicians.

8. Gauss M. Cordeiro (b. 1952)

   - Field: Statistics, probability.  

   - Notable Achievements: Highly cited for his contributions to statistical distributions and regression models, with significant influence in applied mathematics. 

(https://research.com/scientists-rankings/mathematics/br)

   - Why Famous: His prolific research output has made him one of Brazil’s most cited mathematicians.

9. José Mario Martínez (b. 1948)

   - Field: Optimization, numerical analysis.  

   - Notable Achievements**: Known for his work in nonlinear optimization and numerical methods, with applications in computational mathematics.

(https://research.com/scientists-rankings/mathematics/br)

   - Why Famous: His contributions to optimization have practical and academic impact, particularly in Brazil.

10. Luiz Henrique Jacy Monteiro (1918–1975)

    - Field: Algebra, mathematical analysis.  

    - Notable Achievements**: Played a major role in developing Brazilian mathematics in the mid-20th century, particularly through his work at Brazilian universities. 

(https://mathshistory.st-andrews.ac.uk/Biographies/Monteiro/)

    - Why Famous: His foundational contributions helped establish modern mathematics in Brazil.

Notes:  

- This list is curated based on available sources, prioritizing mathematicians with international recognition (e.g., Fields Medal, major awards) or significant influence in Brazil (e.g., educational impact). 

(https://pantheon.world/profile/occupation/mathematician/country/brazil)(https://research.com/scientists-rankings/mathematics/br)(https://proofwiki.org/wiki/Mathematician:Mathematicians/Sorted_By_Nation/Brazil)

- Other notable mathematicians, such as Celso C. Ribeiro, Benar Fux Svaiter, or Welington de Melo, could be included but were less frequently cited in sources for broad fame. 

(https://research.com/scientists-rankings/mathematics/br)(https://www.famousfix.com/list/21st-century-brazilian-mathematicians)

Post and graphic created by Grok.

I apologize for the inaccurate depictions in the title graphic. Below are some photos of the featured mathematicians:

Jacob Palis

(https://upload.wikimedia.org/wikipedia/commons/4/43/Jacob_Palis.jpg)

 Artur  Ávila 

https://commons.wikimedia.org/wiki/File:Artur_%C3%81vila_(cropped).jpg

Júlio César de Melo e Sousa

(https://commons.wikimedia.org/wiki/File:J%C3%BAlio_C%C3%A9sar_de_Melo_e_Sousa.png)

1679 - One important message sent from Earth 31 years ago

In 1974 an interstellar radio transmission was broadcast to the globular cluster Messier 13 from the Arecibo radio telescope in Puerto Rico. The message was 1679 bits long. The length of this message was chosen purposely because 1679 is a semiprime with the prime divisors 23 and 73. The message, when plotted in a 23 x 73 array, produces the graphic shown below:

While the human figure is the most recognized feature in the graphic, there are several other features included in the message (from Cassiday, George. "The Arecibo Message". University of Utah. Archived from the original on October 12, 2013. Retrieved 2013-10-13.):

• The numbers one to ten (white; left to right) 
• The atomic numbers of the elements hydrogen, carbon, nitrogen, oxygen, and phosphorus, which make up deoxyribonucleic acid (DNA) (purple)
• The formulas for the chemical compounds that make up the nucleotides of DNA (green)
• The estimated number of DNA nucleotides in the human genome, and a graphic of the double helix structure of DNA (white and blue, respectively)
• The dimension (physical height, 5'9") of an average man (blue/white), a graphic figure of a human being (red), and the human population of Earth which was about 4 billion at the time (white)
• A graphic of the Solar System (including Pluto), indicating which of the planets the message is coming from (yellow). The Sun is on the left and the third planet, Earth, raised toward the human figure
• A graphic of the Arecibo radio telescope and the dimension (the physical diameter) of the transmitting antenna dish (purple, white, and blue)this message:

https://en.wikipedia.org/wiki/Arecibo_message