Sacks Spiral of Prime Numbers

 

(Image: Claudio Rocchini/Wikimedia Commons CC-BY)

This blog has had prior posts about spirals of prime numbers - see: Prime Number Spirals and Ulam Spiral.

Another variant of the Ulam Spiral is the Sacks Spiral, developed by Robert Sacks in 1994 and is shown above. It is constructed so one perfect square occurs in each full rotation. In the Ulam Spiral, two perfect squares occur in each rotation. 

The graphic was found in an article in Science, 29 July 2024 describing advances in trying to prove the Riemann Hypothesis (one of the remaining unsolved Hilbert Problems).

A374426 Contribution to the OEIS

 This was my 20th sequence published by the OEIS and my 250th contribution.

A374426 a(n) = n*(n + 1)/2 + pi(n), where pi(n) = A000720(n) is the prime counting function. 0

1, 4, 8, 12, 18, 24, 32, 40, 49, 59, 71, 83, 97, 111, 126, 142, 160, 178, 198, 218, 239, 261, 285, 309, 334, 360, 387, 415, 445, 475, 507, 539, 572, 606, 641, 677, 715, 753, 792, 832, 874, 916, 960, 1004, 1049, 1095, 1143, 1191, 1240, 1290, 1341, 1393, 1447 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 1,2 LINKS James C. McMahon, Table of n, a(n) for n = 1..10000 FORMULA a(n) = A000217(n) + A000720(n). a(1) = 1; for n > 1: a(n) = a(n-1) + n + A010051(n). MATHEMATICA Table[n(n+1)/2+PrimePi[n], {n, 53}] CROSSREFS Cf. A000217, A000720, A010051, A256885. Partial sums of A014683. Sequence in context: A311635 A049621 A072511 * A156324 A311636 A311637 Adjacent sequences: A374423 A374424 A374425 * A374427 A374428 A374429 KEYWORD nonn,new AUTHOR James C. McMahon, Jul 08 2024 STATUS approved

Bumping Masses Calculate Pi

(Image: Gemini generated)

Grant Sanderson's site, 3Blue1Brown, has an interesting video about colliding blocks generating the digits of Pi: 6Xh-jhDJ4 .

For an explanation of why this works, see: https://www.youtube.com/watch?v=jsYwFizhncE

He has another video showing another dynamic system using this principle of converting problems of dynamics to problems of geometry: https://www.youtube.com/watch?v=brU5yLm9DZM .

Grant Sanderson's videos are a favorite of this Blog: https://www.youtube.com/@3blue1brown

30,000

 

(Image: Gemini Generated)

As of 7/27/2024, this blog had its 30,000th view. That is certainly not Tik Tok influencer volume, but I'm happy that thousands of visitors have had the opportunity to learn a few new things.

I did a preliminary search for 30,000 to see if there are any interesting facts. Wikipedia has listings for many numbers and it led me to its page where it lists interesting numbers in ranges. For the range 30,000 to 30,999, it listed the following with a few referencing sequences from On-Line Encyclopedia of Integer Sequences (OEIS) - a favorite site of this blog.

• 30029 = primorial prime
• 30030 = primorial^[1]
• 30031 = smallest composite number which is one more than a primorial
• 30203 = safe prime
• 30240 = harmonic divisor number^[2]
• 30323 = Sophie Germain prime and safe prime
• 30420 = pentagonal pyramidal number^[3]
• 30537 = Riordan number
• 30694 = open meandric number
• 30941 = first base 13 repunit prime

https://en.wikipedia.org/wiki/30,000

 1. Sloane, N. J. A. (ed.). "Sequence A002110 (Primorial numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

2. Sloane, N. J. A. (ed.). "Sequence A001599 (Harmonic or Ore numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

 3. Sloane, N. J. A. (ed.). "Sequence A002411 (Pentagonal pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

Top Ten Photographs of the James Webb Space Telescope

Credit: NASA, ESA, CSA, STScI / Image processing by Joseph DePasquale (STScI), Anton M. Koekemoer (STScI), and Alyssa Pagan (STScI)

The James Webb Space Telescope (JWST) was launched December 25, 2021. About six months later, it started producing some of the best images of our universe. PetaPixel recently published the top ten photographs produced so far: https://petapixel.com/2024/07/20/the-james-webb-space-telescopes-10-best-space-photos-so-far/. The iconic Pillars of Creation is shown above. See the the link above for the other top images.

Also see prior post on the JWST: https://jamesmacmath.blogspot.com/2022/01/number-of-planets-in-universe.html

The specialty of the JWST is imaging in the infrared spectrum. NASA has a good article here: https://webb.nasa.gov/content/science/firstLight.html.

Rogue Planets

 

(Image: https://www.iconfinder.com/raminrzdh)

A rogue planet is a planet not orbiting a star. Instead of orbiting stars, rogue planets wander space between the stars. There are estimates that there may be trillions of such planets in our galaxy alone. IEEE Spectrum recently had an article about rogue planets and their detection: https://spectrum.ieee.org/rogue-planet.

Also see: Stars

Black Holes

Planets in the Universe

The James Webb Space Telescope recently found some rogue planets:

https://www.space.com/james-webb-space-telescope-rogue-planets-star-formation

Moving Faster Than the Speed of Light

(Image: https://www.iconfinder.com/ibobicon)

Vsause explores how shadows may appear to travel faster than the speed of light in this YouTube video:

https://www.youtube.com/watch?v=JTvcpdfGUtQ&t=165s

I've had discussions with other people who didn't think it was possible for the shadow to move faster than the speed of light, c. Now I have some back-up material to explain. The key is that although the image appears to be moving faster than c, there is not a physical object moving faster than c, nor is information being transmitted faster than c.

See another post in this blog about a very fast space vehicle: https://jamesmacmath.blogspot.com/2024/01/fast-space-travel.html

A373083 Contribution to the OEIS

 

A373083 a(1) = 1; for n > 1, a(n) = a(n-1) + n if n is prime, else a(n) = a(n-1) + largest divisor of n < n. 0

1, 3, 6, 8, 13, 16, 23, 27, 30, 35, 46, 52, 65, 72, 77, 85, 102, 111, 130, 140, 147, 158, 181, 193, 198, 211, 220, 234, 263, 278, 309, 325, 336, 353, 360, 378, 415, 434, 447, 467, 508, 529, 572, 594, 609, 632, 679, 703, 710, 735, 752, 778, 831, 858, 869, 897 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 1,2 COMMENTS Differs from A219730 at a(12). LINKS James C. McMahon, Table of n, a(n) for n = 1..1000 FORMULA a(n) = a(n-1) + A117818(n) for n > 1. - Michael De Vlieger, Jun 23 2024 EXAMPLE From Michael De Vlieger, Jun 23 2024: (Start) Let lpf = A020639(n). a(2) = 3 since 2 is prime, therefore a(1) + 2 = 3. a(3) = 6 since 3 is prime, therefore a(2) + 3 = 6. a(4) = 8 since 4 is not prime, therefore a(3) + 4/lpf(4) = 6 + 2 = 8. a(5) = 13 since 5 is prime, therefore a(4) + 5 = 13. a(6) = 16 since 6 is not prime, hence a(5) + 6/lpf(6) = 13 + 3 = 16, etc. (End) MATHEMATICA a[1]=1; a[n_]:=If[PrimeQ[n], a[n-1]+n, a[n-1]+Divisors[n][[-2]]]; Table[a[n], {n, 56}] CROSSREFS Cf. A117818, A219730. Sequence in context: A046670 A131383 A219730 * A337484 A139001 A090961 Adjacent sequences: A373080 A373081 A373082 * A373084 A373085 A373086 KEYWORD nonn,easy,new AUTHOR James C. McMahon, Jun 17 2024 STATUS approved

900th Nerdle

This blog previously wrote about the on-line game, Nerdle, which plays similar to the popular word game, Wordle. However, instead of guessing the letters of the word, in Nerdle, one tries to determine the numbers and operations to solve the puzzle.

Today, Nerdle celebrated its 900th puzzle: https://www.nerdlegame.com/900th.html. I encourage you to try it.

Robert Recorde and an Early Multiplication Method

16th century mathematician, Robert Recorde, wrote a books on mathematics. Roman Numerals were used commonly up to his time and he wrote of the advantages of used our current Hindu-Arabic numerals. He introduced a method of multiplication. The method works best for multiplying single-digit numbers, but it also works for higher numbers. 

For example, consider 9 x 7:

Write down the following diagram. The numbers being multiplied are written on the left-hand side of the X; their 10s complements are written on the right-hand side (10-9=1 and 10-7=3):

The right-hand side gives the value of the ones digit of the answer: the product of 1x3. The value of the tens digit is given by the difference of numbers on opposite sides of the X. Both 7-1=6 and 9-3=6, so it doesn't matter which difference is used. Therefore, the final answer is 63 (6x10 +3).

Here is an example with higher numbers. Consider 13 x 7. On the right-hand side of the X, write -3 and 3 (10-13= - 3, and 10-7=3):

The ones unit is -3 x 3 = -9 and the tens digit is 10 which is found by the difference of 13 and 3. So the final answer is 10 tens or 100 -9 = 91.

A little algebra will explain how this X method works. Let the two numbers being multiplied be a and b. The left-hand side of the X becomes a and b, while the right-hand side is 10-a and 10-b.

  

Multiplying the right-hand side of the X, (10-a)x(10-b)=100-10a-10b+ab.

Next take the difference of the opposite sides of the X. This gives us a-(10-b) for the tens unit so we multiply this difference by 10 for 10a-100+10b. Adding this result to the product of the right-hand side gives: 10a-100+10b+100-10a-10b+ab. Note the following terms cancel: 10a & -10a, -100 & 100, and 10b & -10b, leaving the ab as the answer to a times b. 

The math series, Numberphile, recently had a video with Rob Eastaway. He shows one of Recorde's books and walks through the X multiplication method: https://www.youtube.com/watch?v=21Ho32fAEEM.

Also see: Ethiopian Multiplication

2024 First Half - Most Popular Posts

(Image: https://www.iconfinder.com/deemakdaksina)

Below is a list of the top ten posts from this blog for the first half of 2024:

Estimating Populations by the Capture-Recapture Method

Learn how to estimate the population of wildlife species in a given region.

Introduction to Ethiopian Multiplication

This post gives a method of multiplication for any two numbers by simply halving and doubling the numbers.

Pythagorean Presidential Smarts

U.S. 20th President, James A. Garfield developed a proof of the Pythagorean theorem. Also see:

https://jamesmacmath.blogspot.com/2021/09/abraham-lincoln-president-trained-by.html

https://jamesmacmath.blogspot.com/2020/10/a-simple-pythagorean-theorem-proof.html

https://jamesmacmath.blogspot.com/2020/05/proofs-of-pythagorean-theorem.html 

How Rare is that Dollar?

This post will have you examining serial numbers of currency.

My Favorite Math Websites

The title says it all (please comment if you can recommend others).

The Missing Digit Magic Trick

This is an easy-to-perform magic trick based on math. Review the Magic label for additional magic tricks. https://jamesmacmath.blogspot.com/search/label/Magic

Spirals of Prime Numbers

This pattern of prime numbers is attributed to the mathematician Stanislaw Ulam. Also see: https://jamesmacmath.blogspot.com/2020/10/prime-number-spirals.html

Additional Thoughts on the Collatz Conjecture (or the 3n+1 problem)

One of several conjectures discussed in this blog. Also see:

https://jamesmacmath.blogspot.com/2021/12/additional-thoughts-on-goldbach.html

https://jamesmacmath.blogspot.com/2020/04/hotpo-collatz-conjecture.html

https://jamesmacmath.blogspot.com/search/label/Conjectures 

Even Numbers as the Sum of Two Primes 

Another post about a the famous Goldbach conjecture.