Even More Hidden Beast Numbers

In a post titled Hidden Beast Numbers from the 6th August 2023 (Hiroshima Day), I looked at numbers whose sum of prime factors (with and without multiplicity) add to 666. An example would be 998515 where$$ \begin{align} 998515 = 5 \times 7 \times 47 \times 607\\5+7+47+607=666 \end{align}$$In a later post titled More Hidden Beast Numbers from the 21st August 2024, I looked at numbers where the digit sequence 666 shows up when they are rendered in other bases. An example would be 27534 where$$39862_{10}=115666_8$$Today, the number associated with my diurnal age (28330) has the property that$$ \begin{align} 28330 &=2 \times 5 \times 2833 \\ &=5 \times 5666 \end{align}$$I got Gemini to write a program using the following prompt:

Write a program in SageMath that will examine all positive, composite integers in a given range. Let the default range be from 4 to 40000. The criteria to be applied are that the number does not have "6" as a digit and neither do any of its prime factors. However, the prime factors can be combined to form a composite factor containing the digit sequence "666". An example would be 28330 = 2 * 5 * 2833. When factored as 5 * 5666, the factor "5666" contains the digit sequence "666". Output should be a table showing the number, its prime factorisation and its adjusted factorisation. Output should also include a comma-separated list of suitable numbers along with a count of how many there are in the given range.

Here is the output (permalink):

│ Number │ Prime Factorisation  │ Adjusted Factorisation  │

╞════════╪══════════════════════╪═════════════════════════╡

│ 1332   │ 2^2 * 3^2 * 37       │ 2 * 666                 │

├────────┼──────────────────────┼─────────────────────────┤

│ 1998   │ 2 * 3^3 * 37         │ 3 * 666                 │

├────────┼──────────────────────┼─────────────────────────┤

│ 3330   │ 2 * 3^2 * 5 * 37     │ 5 * 666                 │

├────────┼──────────────────────┼─────────────────────────┤

│ 3332   │ 2^2 * 7^2 * 17       │ 2 * 1666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 4998   │ 2 * 3 * 7^2 * 17     │ 3 * 1666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 5328   │ 2^4 * 3^2 * 37       │ 2 * 2 * 2 * 666         │

├────────┼──────────────────────┼─────────────────────────┤

│ 5332   │ 2^2 * 31 * 43        │ 2 * 2666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 5994   │ 2 * 3^4 * 37         │ 3 * 3 * 666             │

├────────┼──────────────────────┼─────────────────────────┤

│ 7332   │ 2^2 * 3 * 13 * 47    │ 2 * 3666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 7992   │ 2^3 * 3^3 * 37       │ 2 * 2 * 3 * 666         │

├────────┼──────────────────────┼─────────────────────────┤

│ 7998   │ 2 * 3 * 31 * 43      │ 3 * 2666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 8330   │ 2 * 5 * 7^2 * 17     │ 5 * 1666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 9324   │ 2^2 * 3^2 * 7 * 37   │ 2 * 7 * 666             │

├────────┼──────────────────────┼─────────────────────────┤

│ 9332   │ 2^2 * 2333           │ 2 * 4666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 9990   │ 2 * 3^3 * 5 * 37     │ 3 * 5 * 666             │

├────────┼──────────────────────┼─────────────────────────┤

│ 10998  │ 2 * 3^2 * 13 * 47    │ 3 * 3666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 11322  │ 2 * 3^2 * 17 * 37    │ 17 * 666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 11332  │ 2^2 * 2833           │ 2 * 5666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 11988  │ 2^2 * 3^4 * 37       │ 2 * 3 * 3 * 666         │

├────────┼──────────────────────┼─────────────────────────┤

│ 13320  │ 2^3 * 3^2 * 5 * 37   │ 2 * 2 * 5 * 666         │

├────────┼──────────────────────┼─────────────────────────┤

│ 13324  │ 2^2 * 3331           │ 2 * 6662                │

├────────┼──────────────────────┼─────────────────────────┤

│ 13328  │ 2^4 * 7^2 * 17       │ 2 * 2 * 2 * 1666        │

├────────┼──────────────────────┼─────────────────────────┤

│ 13330  │ 2 * 5 * 31 * 43      │ 5 * 2666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 13332  │ 2^2 * 3 * 11 * 101   │ 2 * 6666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 13334  │ 2 * 59 * 113         │ 2 * 6667                │

├────────┼──────────────────────┼─────────────────────────┤

│ 13338  │ 2 * 3^3 * 13 * 19    │ 2 * 6669                │

├────────┼──────────────────────┼─────────────────────────┤

│ 13998  │ 2 * 3 * 2333         │ 3 * 4666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 14994  │ 2 * 3^2 * 7^2 * 17   │ 3 * 3 * 1666            │

├────────┼──────────────────────┼─────────────────────────┤

│ 15318  │ 2 * 3^2 * 23 * 37    │ 23 * 666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 15332  │ 2^2 * 3833           │ 2 * 7666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 15984  │ 2^4 * 3^3 * 37       │ 2 * 2 * 2 * 3 * 666     │

├────────┼──────────────────────┼─────────────────────────┤

│ 17982  │ 2 * 3^5 * 37         │ 3 * 3 * 3 * 666         │

├────────┼──────────────────────┼─────────────────────────┤

│ 18330  │ 2 * 3 * 5 * 13 * 47  │ 5 * 3666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 19314  │ 2 * 3^2 * 29 * 37    │ 29 * 666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 19332  │ 2^2 * 3^3 * 179      │ 2 * 9666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 19980  │ 2^2 * 3^3 * 5 * 37   │ 2 * 3 * 5 * 666         │

├────────┼──────────────────────┼─────────────────────────┤

│ 19989  │ 3^2 * 2221           │ 3 * 6663                │

├────────┼──────────────────────┼─────────────────────────┤

│ 19992  │ 2^3 * 3 * 7^2 * 17   │ 2 * 2 * 3 * 1666        │

├────────┼──────────────────────┼─────────────────────────┤

│ 19995  │ 3 * 5 * 31 * 43      │ 3 * 6665                │

├────────┼──────────────────────┼─────────────────────────┤

│ 19998  │ 2 * 3^2 * 11 * 101   │ 3 * 6666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 20001  │ 3 * 59 * 113         │ 3 * 6667                │

├────────┼──────────────────────┼─────────────────────────┤

│ 20007  │ 3^4 * 13 * 19        │ 3 * 6669                │

├────────┼──────────────────────┼─────────────────────────┤

│ 21312  │ 2^6 * 3^2 * 37       │ 2 * 2 * 2 * 2 * 2 * 666 │

├────────┼──────────────────────┼─────────────────────────┤

│ 21328  │ 2^4 * 31 * 43        │ 2 * 2 * 2 * 2666        │

├────────┼──────────────────────┼─────────────────────────┤

│ 21332  │ 2^2 * 5333           │ 2 * 10666               │

├────────┼──────────────────────┼─────────────────────────┤

│ 21978  │ 2 * 3^3 * 11 * 37    │ 3 * 11 * 666            │

├────────┼──────────────────────┼─────────────────────────┤

│ 22998  │ 2 * 3 * 3833         │ 3 * 7666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 23310  │ 2 * 3^2 * 5 * 7 * 37 │ 5 * 7 * 666             │

├────────┼──────────────────────┼─────────────────────────┤

│ 23324  │ 2^2 * 7^3 * 17       │ 2 * 7 * 1666            │

├────────┼──────────────────────┼─────────────────────────┤

│ 23330  │ 2 * 5 * 2333         │ 5 * 4666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 23332  │ 2^2 * 19 * 307       │ 2 * 11666               │

├────────┼──────────────────────┼─────────────────────────┤

│ 23994  │ 2 * 3^2 * 31 * 43    │ 3 * 3 * 2666            │

├────────┼──────────────────────┼─────────────────────────┤

│ 24990  │ 2 * 3 * 5 * 7^2 * 17 │ 3 * 5 * 1666            │

├────────┼──────────────────────┼─────────────────────────┤

│ 25308  │ 2^2 * 3^2 * 19 * 37  │ 2 * 19 * 666            │

├────────┼──────────────────────┼─────────────────────────┤

│ 25332  │ 2^2 * 3 * 2111       │ 2 * 12666               │

├────────┼──────────────────────┼─────────────────────────┤

│ 25974  │ 2 * 3^3 * 13 * 37    │ 3 * 13 * 666            │

├────────┼──────────────────────┼─────────────────────────┤

│ 27972  │ 2^2 * 3^3 * 7 * 37   │ 2 * 3 * 7 * 666         │

├────────┼──────────────────────┼─────────────────────────┤

│ 28322  │ 2 * 7^2 * 17^2       │ 17 * 1666               │

├────────┼──────────────────────┼─────────────────────────┤

│ 28330  │ 2 * 5 * 2833         │ 5 * 5666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 28998  │ 2 * 3^4 * 179        │ 3 * 9666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 29304  │ 2^3 * 3^2 * 11 * 37  │ 2 * 2 * 11 * 666        │

├────────┼──────────────────────┼─────────────────────────┤

│ 29328  │ 2^4 * 3 * 13 * 47    │ 2 * 2 * 2 * 3666        │

├────────┼──────────────────────┼─────────────────────────┤

│ 29332  │ 2^2 * 7333           │ 2 * 14666               │

├────────┼──────────────────────┼─────────────────────────┤

│ 29970  │ 2 * 3^4 * 5 * 37     │ 3 * 3 * 5 * 666         │

├────────┼──────────────────────┼─────────────────────────┤

│ 29988  │ 2^2 * 3^2 * 7^2 * 17 │ 2 * 3 * 3 * 1666        │

├────────┼──────────────────────┼─────────────────────────┤

│ 31302  │ 2 * 3^2 * 37 * 47    │ 47 * 666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 31332  │ 2^2 * 3 * 7 * 373    │ 2 * 15666               │

├────────┼──────────────────────┼─────────────────────────┤

│ 31992  │ 2^3 * 3 * 31 * 43    │ 2 * 2 * 3 * 2666        │

├────────┼──────────────────────┼─────────────────────────┤

│ 31998  │ 2 * 3 * 5333         │ 3 * 10666               │

├────────┼──────────────────────┼─────────────────────────┤

│ 32994  │ 2 * 3^3 * 13 * 47    │ 3 * 3 * 3666            │

├────────┼──────────────────────┼─────────────────────────┤

│ 33300  │ 2^2 * 3^2 * 5^2 * 37 │ 2 * 5 * 5 * 666         │

├────────┼──────────────────────┼─────────────────────────┤

│ 33310  │ 2 * 5 * 3331         │ 5 * 6662                │

├────────┼──────────────────────┼─────────────────────────┤

│ 33315  │ 3 * 5 * 2221         │ 5 * 6663                │

├────────┼──────────────────────┼─────────────────────────┤

│ 33320  │ 2^3 * 5 * 7^2 * 17   │ 2 * 2 * 5 * 1666        │

├────────┼──────────────────────┼─────────────────────────┤

│ 33324  │ 2^2 * 3 * 2777       │ 2 * 16662               │

├────────┼──────────────────────┼─────────────────────────┤

│ 33325  │ 5^2 * 31 * 43        │ 5 * 6665                │

├────────┼──────────────────────┼─────────────────────────┤

│ 33328  │ 2^4 * 2083           │ 2 * 16664               │

├────────┼──────────────────────┼─────────────────────────┤

│ 33330  │ 2 * 3 * 5 * 11 * 101 │ 5 * 6666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 33334  │ 2 * 7 * 2381         │ 2 * 16667               │

├────────┼──────────────────────┼─────────────────────────┤

│ 33335  │ 5 * 59 * 113         │ 5 * 6667                │

├────────┼──────────────────────┼─────────────────────────┤

│ 33338  │ 2 * 79 * 211         │ 2 * 16669               │

├────────┼──────────────────────┼─────────────────────────┤

│ 33345  │ 3^3 * 5 * 13 * 19    │ 5 * 6669                │

├────────┼──────────────────────┼─────────────────────────┤

│ 34998  │ 2 * 3 * 19 * 307     │ 3 * 11666               │

├────────┼──────────────────────┼─────────────────────────┤

│ 35298  │ 2 * 3^2 * 37 * 53    │ 53 * 666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 35332  │ 2^2 * 11^2 * 73      │ 2 * 17666               │

├────────┼──────────────────────┼─────────────────────────┤

│ 37324  │ 2^2 * 7 * 31 * 43    │ 2 * 7 * 2666            │

├────────┼──────────────────────┼─────────────────────────┤

│ 37328  │ 2^4 * 2333           │ 2 * 2 * 2 * 4666        │

├────────┼──────────────────────┼─────────────────────────┤

│ 37998  │ 2 * 3^2 * 2111       │ 3 * 12666               │

├────────┼──────────────────────┼─────────────────────────┤

│ 38318  │ 2 * 7^2 * 17 * 23    │ 23 * 1666               │

├────────┼──────────────────────┼─────────────────────────┤

│ 38330  │ 2 * 5 * 3833         │ 5 * 7666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 39294  │ 2 * 3^2 * 37 * 59    │ 59 * 666                │

├────────┼──────────────────────┼─────────────────────────┤

│ 39332  │ 2^2 * 9833           │ 2 * 19666               │

├────────┼──────────────────────┼─────────────────────────┤

│ 39972  │ 2^2 * 3 * 3331       │ 2 * 3 * 6662            │

├────────┼──────────────────────┼─────────────────────────┤

│ 39978  │ 2 * 3^2 * 2221       │ 2 * 3 * 6663            │

├────────┼──────────────────────┼─────────────────────────┤

│ 39984  │ 2^4 * 3 * 7^2 * 17   │ 2 * 2 * 2 * 3 * 1666    │

├────────┼──────────────────────┼─────────────────────────┤

│ 39990  │ 2 * 3 * 5 * 31 * 43  │ 3 * 5 * 2666            │

└────────┴──────────────────────┴─────────────────────────┘

Suitable numbers: 

 

1332, 1998, 3330, 3332, 4998, 5328, 5332, 5994, 7332, 7992, 7998, 8330, 9324, 9332, 9990, 10998, 11322, 11332, 11988, 13320, 13324, 13328, 13330, 13332, 13334, 13338, 13998, 14994, 15318, 15332, 15984, 17982, 18330, 19314, 19332, 19980, 19989, 19992, 19995, 19998, 20001, 20007, 21312, 21328, 21332, 21978, 22998, 23310, 23324, 23330, 23332, 23994, 24990, 25308, 25332, 25974, 27972, 28322, 28330, 28998, 29304, 29328, 29332, 29970, 29988, 31302, 31332, 31992, 31998, 32994, 33300, 33310, 33315, 33320, 33324, 33325, 33328, 33330, 33334, 33335, 33338, 33345, 34998, 35298, 35332, 37324, 37328, 37998, 38318, 38330, 39294, 39332, 39972, 39978, 39984, 39990 

 

Total count in range [4, 40000]: 96

Truncated Tetrahedral Numbers

The number associated with my diurnal age today, 28325, has a property that allows it membership of OEIS A100156:

 A100156: structured truncated tetrahedral numbers.

So what is a truncated tetrahedron. It belongs to a class of three-dimensional figurate numbers. They are geometrically constructed by taking a regular tetrahedral number of a given size and removing smaller tetrahedral corners from each of its four vertices.

What does it look like? Figure 1 shows an example.

Figure 1: source

Figure 2 shows the associated net made up from four hexagons and four equilateral triangles.

Figure 2

The details are:

Faces: 8

Edges: 18

Vertices: 12 

A structured truncated tetrahedral number (OEIS A100156) is a three-dimensional figurate number that counts the total number of points (or spheres) in a truncated tetrahedron built outward in nested shells from a single corner vertex, where every polygonal face is populated as a classical vertex-grown figurate polygon rather than a centered polygon.

The initial members of the sequence are: 1, 12, 44, 108, 215, 376, 602, 904, 1293, 1780, 2376, 3092, 3939, 4928, 6070, 7376, 8857, 10524, 12388, 14460, 16751, 19272, 22034, 25048, 28325, 31876, 35712, 39844

 

Figure 3

To quote from Gemini:

The shape first emerged prominently in the 15th and 16th centuries as artists sought to calculate and render perfect three-dimensional forms on flat surfaces:

Leonardo da Vinci & Luca Pacioli (1509): The most celebrated early appearance of the truncated tetrahedron is in Fra Luca Pacioli's treatise, De Divina Proportione. Leonardo da Vinci provided the illustrations, drawing the solid in two revolutionary forms: solidus (a solid form showing surface planes) and vacuus (a skeletal wireframe showing its interior framework). See Figure 3.

Albrecht Dürer (1525): In his foundational work Underweysung der Messung (Instruction in Measurement), German master Albrecht Dürer revolutionized geometric study by publishing the first foldable paper nets. His flat layout of the truncated tetrahedron allowed readers to cut out and construct physical models, making complex perspective easier to perceive in physical space.

Even Number Bias

 I asked Gemini the following:

Whenever I set the temperature of the air conditioner, I invariably choose an even number (usually 24) and seldom do I choose an odd number. I only realised this when I’d accidentally set it to 23 degrees Celsius. Is there any research into the possibility of an even number bias whenever there is a choice of even or odd numbers?

Here is its response:

There is a robust body of cognitive psychology and numerical cognition research confirming that humans exhibit a distinct bias toward even numbers when setting arbitrary values like thermostat temperatures, television volumes, and alarm clocks. In cognitive science, this phenomenon is driven by processing fluency, the MARC effect, and semantic associations tied to parity.

1. The Parity Effect and Processing Fluency

In the study of numerical cognition, researchers have long observed that the human brain does not treat odd and even numbers equally. Known as the odd-even effect or parity preference, studies consistently demonstrate that people process even numbers faster and with fewer errors than odd numbers.

• Reaction Time Studies: In classic parity judgment tasks (where subjects must quickly press a button to identify whether a flashed digit is odd or even), Terence Hines (1990) and subsequent researchers found that human reaction times are significantly faster for even digits than for odd ones.

• Addition and Multiplication Fluency: Because we learn our multiplication tables early in life, and because multiplying any integer by an even number always yields an even result, even numbers occur much more frequently in our mental arithmetic lexicon.

• Cognitive Ease: Psychologists refer to this speed and familiarity as processing fluency. When our brains can process a stimulus with less friction, we experience a subtle, subconscious positive affect toward it. When you look at a thermostat display, 24 simply "feels" more settled and harmonious than 23 because your brain processes it with slightly less cognitive effort.

2. Divisibility and the Need for Symmetry

From a mathematical standpoint, the defining characteristic of an even number is its divisibility by 2 without a remainder. In everyday psychology, this translates into a visceral sense of balance, symmetry, and completeness.

• Mental Halving: Even numbers can be split cleanly into two equal parts (24 = 12 + 12), whereas odd numbers leave a " messy" remainder or fraction (23 = 11.5 + 11.5). When setting an environmental control like an air conditioner or a car radio volume, the brain treats the number not just as a measurement of heat or sound, but as a visual and conceptual anchor. An odd number like 23 subconsciously registers as "incomplete" or "off-balance," sitting uneasily between two stable points (22 and 24).

• The "Round Number" Heuristic: Research by Schindler and Kirby (1997) into pricing and numerical preferences shows that humans gravitate toward numbers that feel "round" or serve as natural cognitive reference points. While multiples of 5 and 10 are the strongest anchors (which is why many people tolerate 25 on a volume dial even though it is odd), even numbers act as secondary anchors of stability across the entire number line.

3. Semantic Associations: "Good" vs. "Strange"

Psychologist Laurence E. Bourne Jr. and colleagues (such as Wilkie & Bodenhausen, 2012) have investigated the deep-seated semantic and linguistic associations tied to parity. Across many cultures and languages, the very words used to describe parity carry heavy emotional and aesthetic weight:

• Linguistic Framing: In English, "even" is synonymous with calm, balanced, fair, smooth, and level-headed. Conversely, "odd" literally means strange, peculiar, mismatched, deviant, or left over.

• Implicit Personality and Gender Attribution: In experiments where participants were asked to rate the abstract "personalities" of digits, even numbers were consistently rated as more agreeable, peaceful, and comforting, while odd numbers were rated as more aggressive, jagged, and active. When your goal is to create a comfortable, climate-controlled environment, your subconscious naturally aligns the desired physical state (calm, balanced comfort) with the numerical category that shares those exact semantic traits.

4. The MARC Effect (Spatial-Numerical Association)

Another neurological quirk discovered by Jean-Paul Nuerk and colleagues (2004) is the MARC effect (Markedness Association of Response Codes).

• Linguistic Markedness: In linguistics, pairs of opposites usually have an "unmarked" (default, positive, primary) term and a "marked" (secondary, negated, or derivative) term—such as tall/short or happy/unhappy. In numerical cognition, even is the unmarked default state, while odd is the marked deviation.

• Right-Side Advantage: For right-handed individuals (and most left-handed individuals due to reading direction), the brain spatially maps "even" to the right side of space—the dominant, preferred side for action—and "odd" to the left. Because we associate our dominant side with fluency and control, choosing an even number on a digital interface subtly reinforces a sense of mastery and order over our surroundings.

Why 24 Specifically Stands Out

Your specific preference for 24 over other numbers in that temperature range has an extra layer of mathematical reinforcement. Beyond simply being an even number in the ideal thermal comfort zone for tropical and subtropical climates, 24 is a highly composite number.

Whereas 22 is only divisible by 1, 2, 11, and 22, and 26 is only divisible by 1, 2, 13, and 26, the number 24 has eight factors (1, 2, 3, 4, 6, 8, 12, 24). In numerical cognition, numbers with high divisibility (like 12, 24, 36, and 60) possess the highest degree of mental fluency because we encounter them constantly in timekeeping (24 hours in a day), geometry, and measurement. Spotting 23 on your display jarred your attention because you moved from one of the most divisible, harmonious integers in daily life to a prime number with no factors other than 1 and itself. 

Embedded Primes Versus Prime Partitions

I noticed that the number associated with my diurnal age today (28323), while not prime, has digits that are all prime with the exception of 8 and it also has two other primes contained within it (283 and 23). I asked Gemini the following question:

28323 is a composite number with the property that it is a concatenation of prime numbers in various ways e.g. 283 || 23 and 2 || 83 || 23 where I’ve used the symbol || for concatenation. Write a program in SageMath that will identify all composite numbers in a given range that are concatenations of prime numbers in two or more different ways. The output should be a list of comma-separated numbers. The default range can be from 4 to 40000 and the upper and lower limits of the range should be easily adjustable. There should also be a switch so that the program can search for prime numbers with the same property, although the default should be set to composite. The number of suitable concatenations should be adjustable as well, although 2 is to be the default. Can a count of the suitable numbers be included in the output, underneath the list of numbers.

The program as requested is quite flexible. Below are the opening lines of code:

# --- Adjustable Parameters ---

LOWER_LIMIT = 4

UPPER_LIMIT = 40000

SEARCH_PRIMES = True  # False = search for composite numbers (default), True = search for primes

MIN_WAYS = 11         # Minimum number of valid prime concatenations required

# -----------------------------

Using these settings and thus searching for all primes in the range from 4 to 40000 that can be concatenated in at least 11 different ways, I found only a single number: 37337 (permalink). However, OEIS A179919 that lists primes with eleven embedded primes listed 111317 as the first number. Clearly I was confused and I sought clarification from Gemini. Here is its explanation:

The discrepancy comes down to two fundamentally different mathematical definitions: full-string prime partitions (concatenations) versus distinct proper prime substrings (embedded primes).

Your SageMath program counts the number of ways an entire number can be sliced into a complete concatenation of prime pieces from left to right, whereas OEIS A179919 counts how many distinct prime numbers appear anywhere inside the number as contiguous substrings (excluding the prime itself).

1. What Your Program Counts: Prime Concatenations (37337)

Your program requires every single digit of the number to be used in each partition. A 5-digit number has 2^{5-1} - 1 = 15 possible ways to be sliced into two or more contiguous pieces.

For the prime number 37337, 11 of those 15 full-string partitions consist entirely of prime numbers:

1. 3 || 7 || 3 || 3 || 7

2. 3 || 7 || 3 || 37

3. 3 || 7 || 337

4. 3 || 73 || 3 || 7

5. 3 || 73 || 37

6. 37 || 3 || 3 || 7

7. 37 || 3 || 37

8. 37 || 337

9. 373 || 3 || 7

10. 373 || 37

11. 3733 || 7

(The only 4 slices of 37337 that fail are the ones containing the composite substrings 33 or 733: 3 || 7 || 33 || 7, 3 || 733 || 7, 37 || 33 || 7, and 3 || 7337.)

2. What OEIS A179919 Counts: Distinct Embedded Primes (111317)

In the OEIS family of sequences (such as A039996 and A179909–A179924), an "embedded prime" is defined as a distinct prime proper substring occurring anywhere within the base-10 representation of the prime.

Under the OEIS definition, three completely different rules apply:

• The rest of the digits do not need to form primes: You just extract a contiguous block of digits from the number and ignore the digits to the left and right.

• Duplicates are only counted once: You collect the substrings into a mathematical set of unique prime values.

• Composite digits like 1 are helpful: Because you don't have to partition the entire number into primes, having the digit 1 (which is not prime and ruins full concatenations) is actually an advantage in A179919 because 1 starts many multi-digit primes (11, 13, 17, 113, 131, 317, etc.).

If we evaluate your number 37337 using the OEIS "embedded primes" rule, it only has 6 embedded primes:

$$\{3, 7, 37, 73, 337, 373, 3733\} \longrightarrow \text{7 primes?}$$

Actually, checking all contiguous proper substrings of 37337:

• 1-digit: 3, 7 (repeats ignored)

• 2-digit: 37, 73 (33 is composite)

• 3-digit: 373, 337 (733 is prime! Wait: 733 is indeed prime!)

• 4-digit: 3733 (prime), 7337 (7337 = 11 \times 667, composite)

Even with 733 being prime, the set of distinct proper prime substrings of 37337 is \{3, 7, 37, 73, 337, 373, 733, 3733\}, which is only 8 distinct embedded primes (falling short of 11). Furthermore, note that because 733 is prime, 37337 actually has 12 valid prime concatenations in your program (3 || 733 || 7 is also valid!), which is why it passed your MIN_WAYS = 11 threshold (>= 11).

Now compare that to 111317, the first term of OEIS A179919:

• Under your concatenation program: 111317 has only 1 valid prime concatenation (11 || 13 || 17), because any slice that isolates a single 1 (1 || 11317, 111 || 3 || 17, etc.) fails since 1 is not prime.

• Under OEIS A179919 (distinct proper prime substrings): Scanning all contiguous substrings of 111317 yields 11 distinct primes:

  1. 1-digit: 3, 7 (2 primes)

  2. 2-digit: 11, 13, 17, 31 (4 primes)

  3. 3-digit: 113, 131, 317 (3 primes)

  4. 4-digit: 1113 (3 \times 371, composite), 1131 (3 \times 377, composite), 1317 (3 \times 439, composite) (0 primes)

  5. 5-digit: 11131 (prime), 11317 (prime) (2 primes)

Summing those unique prime substrings gives 2 + 4 + 3 + 0 + 2 = 11 embedded primes:

$$\{3, 7, 11, 13, 17, 31, 113, 131, 317, 11131, 11317\}$$

Side-by-Side Comparison of the Two Rules

Property	Your Program (Prime Concatenations)	OEIS A179919 (Embedded Primes)
Core Mechanism	Partitions the entire string into adjacent prime blocks (A || B || C).	Extracts any contiguous prime substring from anywhere inside the string.
Unused Digits	Forbidden — every digit must belong to a prime block in the partition.	Allowed — surrounding digits outside the substring are ignored.
Repeated Primes	Counted every time they form a distinct full-string partition.	Ignored — only distinct prime values are counted.
Score for 37337	12 valid full-string prime concatenations.	8 distinct embedded primes (\{3, 7, 37, 73, 337, 373, 733, 3733\}).
Score for 111317	1 valid full-string prime concatenation (11 || 13 || 17).	11 distinct embedded primes (\{3, 7, 11, 13, 17, 31, 113, 131, 317, 11131, 11317\}).

I've gotten Gemini to write a program that accepts any positive integer as input, prime or non-prime, and outputs the concatenated primes as well as the embedded primes. Here is the output for 28323 (permalink):

Analysis for 28323 (Composite)

=============================================

1. PRIME CONCATENATIONS (Total: 4)

   1. 2 || 83 || 2 || 3

   2. 2 || 83 || 23

   3. 283 || 2 || 3

   4. 283 || 23

2. DISTINCT EMBEDDED PRIMES (Total: 5)

   2, 3, 23, 83, 283

I've added this program to my daily number analysis.

More on Days and Dates

In two recent posts, When Days and Dates Align (8th October 2026) and Numbers as Dates (29th April 2026), I've treated my diurnal age's connection with dates using a DD-M-YY or D-M-YY format with the YY being mandatory though ambiguous. As an example, my diurnal age today is 28322 and so in this scheme would be interpreted as 28th March 2022 if we were looking for a date closest to the current date of 18th October 2026).

Another option is to consider a YY-M-DD or YY-MM-D format. Using this format the year changes slowly and today's date would be considered to be 22nd March 2028. This is probably the better format and using this I posed Gemini the following problem:

Today I’m 28322 days old counting my birthday on April 3rd 1949 as Day 0. Interpreting 28322 as YY-M-DD format gives 22nd March 1928. Not 22nd March 2028 because that date is in the future. As my diurnal age increases by 1 each day, when will I reach a number that, interpreted in YY-M-DD or YY-MM-D format, will be in the past. I want the number and its associated date.

Here was its response:

The number is \( \textbf{29201}\), and you will reach it on \( \textbf{15 March 2029}\).

When you are 29,201 days old on 15 March 2029, hyphenating the number in YY-M-DD format gives 29-2-01, which is 1 February 2029—a 21st-century date that is 42 days in the past at that moment.

Here is why no earlier number works:

• 28801 to 28930 (9 February – 17 June 2028): These are the only valid dates in the 28xxx series left, and they represent August and September 2028—which are still months in the future when you reach those ages.

• 29000 (26 August 2028): Your day count rolls into the 29xxx series while it is still 2028, so 29 remains in the future for the rest of that year.

• 29101 to 29131 (5 December 2028 – 4 January 2029): Even when 2029 begins on 1 January 2029 (when you are 29128 days old), 29-1-28 (28 January 2029) and 29-12-8 (8 December 2029) are still ahead of you, as are 29129, 29130, and 29131.

• 29132 to 29200 (5 January – 14 March 2029): None of these 69 numbers can form a valid calendar date in either YY-M-DD or YY-MM-D format.

• 29201 (15 March 2029): The very first valid date after that gap is 29-2-01 (1 February 2029), making it the first 21st-century date that lies in the past.

All of this is just some silliness to entertain myself. 

Double-Safe and Triple-Safe Primes

A safe prime \(p\) is one such that \( \dfrac{p-1}{2} \) is also prime.

The first example of such a prime is 5 because \( \dfrac{5-1}{2} =2\) which is prime.

However, we can generalise this notion and speak of a \(k\)-safe prime \(p\) where $$ \frac{p-(2^i-1)}{2^i} \text{ is prime for } i=0,1, 2, \dots  k$$Let's look at the various values of \(k\) as applied to a prime \(p\) :

• \(i=0\) just gives the originating prime \(p\)

• \(i=1\) means the prime is safe and  \( \dfrac{p-1}{2} \) is prime

• \(i=2\) means the prime is 2-safe or double-safe and  \( \dfrac{p-3}{4} \) is prime

• \(i=3\) means the prime is 3-safe or triple-safe and  \( \dfrac{p-7}{8} \) is prime

Let's look at a specific prime, 28319, that is known be a triple safe prime:

\(28319\) is prime

\(28319 \rightarrow \dfrac{28319-1}{2}  =14159\) which is thus a safe prime

\(28319 \rightarrow \dfrac{28319-3}{4} =7079\) which is thus a double-safe prime

\(28319 \rightarrow \dfrac{28319-7}{8}  =3539\) which is thus a triple-safe prime

\(28319 \rightarrow  \dfrac{28319-15}{16} =1769= 29 \times 61 \) which is not prime

These triple-safe primes form OEIS A157358 and the initial members are: 

23, 47, 719, 1439, 2879, 4079, 9839, 11279, 21599, 28319, 51599, 84719, 92399, 95279, 96959, 137279, 157679, 159119, 178799, 209519, 219839, 243119, 349199, 429119, 430799, 441839, 462719, 481199, 491279, 507359, 533999, 571199, 597599

As can be seen, 28319 is the last triple-safe prime that I'll see in my lifetime. The double-safe primes form OEIS A066179 and these are more frequent in the range up to 40000:

11, 23, 47, 167, 359, 719, 1439, 2039, 2879, 4079, 4127, 4919, 5639, 5807, 5927, 6047, 7247, 7559, 7607, 7727, 9839, 10799, 11279, 13799, 13967, 14159, 15287, 15647, 20327, 21599, 21767, 23399, 24407, 24527, 25799, 28319, 28607, 29399, 31607, 35879, 36887

What about quadruple, quintuple-safe primes and beyond. Up to 40000, there are only three quadruple-safe primes and these are 47, 1439 and 2879. Of these, only 2879 is a quintuple-safe prime (permalink). There is no sextuple-safe prime until we reach 71850239! Up to a billion, there are no septuple-safe primes.

The Antisigma Function Revisited

The day after my birthday this year I made a post titled The Antisigma Function. This function returns the sum of the non-divisors of a number and is given by the formula:$$ \text{antisigma}(n)=\frac{n(n+1)}{2}-\sigma(n)$$I noticed today that the number associated with my diurnal age today (28318) has an antisigma value given by:$$ \begin{align} \text{antisigma}(28318) &= 28318 \times 28319/2-\sigma(28318)\\ &=400968721-42480\\ &=400926241 \end{align}$$This antisigma value is almost a multiple of 28318 since:$$ 400926241\!  \! \! \mod 28318 =28315$$This got me wondering whether there are numbers whose antisigma values are multiples of itself. I asked Gemini to investigate and it came to the conclusion that there are no composite integers up to 40,000 (and likely no integers at all) that satisfy this condition. End of story, right. Actually, no. Using the program that Gemini provided I found the following numbers that satisfy (permalink).

--- Results for Composite Integers 1 to 40000 ---

Table:

Composite Integer (n)     | Antisigma Value     

------------------------------------------------

24                        | 240                 

4320                      | 9318240             

4680                      | 10937160            

26208                     | 343351008            

Comma-separated list of the integers:

24, 4320, 4680, 26208 

Gemini had to admit its error. So never trust anything Gemini tells you straight up. I then asked if these were the only numbers and it turns out there are infinitely many more. Here is the explanation that Gemini provided:

For an integer $n$ to divide its antisigma value perfectly, the antisigma expression modulo $n$ must equal 0:

$$\frac{n(n+1)}{2} - \sigma(n) \equiv 0 \pmod n$$

For any even number $n$, the formula for the sum of all integers up to $n$ expands to:

$$\frac{n(n+1)}{2} = \frac{n^2}{2} + \frac{n}{2} = n\left(\frac{n}{2}\right) + \frac{n}{2}$$

When this triangle number sum is divided by $n$, it consistently leaves a remainder of $\frac{n}{2}$. For the entire antisigma value to be perfectly divisible by $n$, the divisor sum $\sigma(n)$ must also leave a remainder of $\frac{n}{2}$ to exactly cancel out the first remainder. Algebraically, this requires:

$$\sigma(n) = kn + \frac{n}{2}$$

Dividing both sides by $n$ reveals the core ratio:

$$\frac{\sigma(n)}{n} = k + 0.5$$

Numbers whose abundancy index (the sum of their divisors divided by the number itself) equals a half-integer (such as 1.5, 2.5, or 3.5) belong to a specific mathematical class known as hemiperfect numbers.

The Online Encyclopedia of Integer Sequences catalogs these half-integral abundancy numbers under OEIS A159907. The first five terms in this sequence are 2, 24, 4320, 4680, and 26208.

The sequence does not end there. It continues into much larger integers that satisfy the exact same divisibility condition:

• 8,910,720
• 17,428,320
• 20,427,264
• 91,963,648
• 197,064,960

Because there is a massive mathematical "desert" spanning from 26,208 all the way to 8.9 million, a programmatic search capped at a 40,000 limit naturally halts after finding 26,208. Therefore, 24, 4320, 4680, and 26208 are not the only composite integers that divide their antisigma values evenly; they are simply the first four composite entries in a much larger, ongoing sequence.

************************

I wrote about hemiperfect numbers in a post titled Hemiperfect Numbers on the 3rd of January 2021.