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

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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