Reversible Sphenic Numbers

There is the reversible prime, known as an emirp. There is the reversible semiprime, known as an emirpimes, and then there is the reversible sphenic number, known as an cinehps. This is a rather ugly term so I'll just use the term reversible sphenic number. An example of an emirp is 17 whose reversal, 71, is also prime. An example of an emirpimes is 26 = 2 x 13 whose reversal, 62 = 2 x 31, is also a semiprime. The first example of a cinehps is 165 = 3 x 5 x 11 whose reversal, 561 = 3 x 11 x 17, is also a sphenic number.

These numbers form OEIS A270175:

A270175

Cinehps numbers: sphenic numbers whose reversal is a different sphenic number.

Note that palindromic sphenic numbers are excluded. The initial members of the sequence are:

165, 246, 285, 286, 366, 418, 435, 438, 498, 534, 561, 582, 609, 642, 663, 682, 759, 814, 834, 894, 906, 957, 1002, 1023, 1034, 1066, 1095, 1113, 1131, 1185, 1209, 1239, 1245, 1265, 1311, 1342, 1353, 1374, 1398, 1419, 1443, 1446, 1479, 1515, 1526, 1542, 1545, ...

Up to the one million mark, these numbers total 5.28% of the range. One of the concepts associated with a sphenic number is that of the sphenic brick. Let's consider a sphenic number \(n\) whose factors are \(a,b,c\). The sphenic brick is the three dimensional cuboid with volume of \(n\) cubic units and linear dimensions of \(a,b\) and \(c\) units.

Such a brick has an associated surface area and a thought occurred to me. Are there any reversible sphenic number pairs that each have the same surface area? A little investigation revealed that there are. They are rare birds indeed however, and there are only eight ( in four pairs) in the range up to three million (366 and 663, 3245 and 5423, 3685 and 5863, 921239 and 932129). Here are the details and here is the permalink to the calculation. Surface areas are shown in bold red.

366  = 2 x 3 x 61 --> 622 and 663 = 3 x 13 x 17 --> 622

3245 = 5 x 11 x 59 --> 1998 and 5423 = 11 x 17 x 29 --> 1998

3685 = 5 x 11 x 67 --> 2254 and 5863 = 11 x 13 x 41 --> 2254

921239 = 11 x 89 x 941 --> 190158 and 932129 = 11 x 101 x 839 --> 190158

None of the sphenic bricks associated with these numbers look like bricks because they are all very elongated but that's the term that is used for these shapes. Figure 1 shows that the 11 x 89 x 941 looks more like a plank than a brick.

Figure 1

There may well be more beyond the three million mark but SageMathCell timed out above that. Anyway, fascinating that such numbers exist with the first of them being 366, the number of days in a leap year. It can be noted that 366 is the only even number. Placed in sequence on the number line we have:

366, 663, 3245, 3685, 5423, 5863, 921239, 932129

This sequence of terms could be described thus:

Non-palindromic sphenic numbers which, when reversed, are also sphenic numbers with the members of both pairs having identical sphenic brick surface areas. 

Not surprisingly this sequence does not appear in the OEIS, nor will it, as I have ceased to contribute.

ADDENDUM:

The above idea can be applied to semiprimes as well. In this case we will be working with two dimensional rectangles. The two factors of the semiprime form the length and width of the associated rectangle. In the range up to one million, reversible semiprimes comprise 6.06% of the total range but there are only 26 reversible semiprimes with the property that their associated areas are equal. See permalink for calculation.

The pairs (up to one million) are:

14269 = 19 x 751 --> 1540 and 96241 = 157 x 613 --> 1540

15167 = 29 x 523 --> 1104 and 76151 = 271 x 281 --> 1104

16237 = 13 x 1249 --> 2524 and 73261 = 61 x 1201 --> 2524

18449 = 19 x 971 --> 1980 and 94481 = 107 x 883 --> 1980

18977 = 7 x 2711 --> 5436 and 77981 = 29 x 2689 --> 5436

36679 = 43 x 853 --> 1792 and 97663 = 127 x 769 --> 1792

140941 = 97 x 1453 --> 3100 and 149041 = 103 x 1447 --> 3100

150251 = 347 x* 433 --> 1560 and 152051 = 383 x 397 --> 1560

196891 = 401 x 491 --> 1784 and 198691 = 431 x 461 --> 1784

302363 = 211 x 1433 --> 3288 and 363203 = 263 x 1381 --> 3288

308459 = 173 x 1783 --> 3912 and 954803 = 937 x 1019 --> 3912

319853 = 317 x 1009 --> 2652 and 358913 = 379 x 947 --> 2652

958099 = 761 x 1259 --> 4040 and 990859 = 839 x 1181 --> 4040

It can be noted that all these reversible semiprimes are odd.

If we are even more adventurous we can consider numbers with four distinct factors as being associated with 4-dimensional hypercuboids. When unfolded into 3-dimensions, the eight cuboids are the equivalent of the surface area of our sphenic bricks. In the range up to three million, about 1.48% of numbers are reversible four factor numbers with the first one being 1518 = 2 x 3 x 11 x 23 and 8151 = 3 x 11 x 13 x 19. However, no number pairs emerge with identical volumes of unfolded cuboids. Figure 1 shows an unfolded hypercube and an unfolded hypercuboid will look similar but will have four pairs of cuboids each with different volumes, instead of consisting of eight identical cubes. Permalink (which may need double checking).

Figure 1

Visualisation of Semiprimes

I've written previously about semiprimes in varying contexts. These posts are listed below:

• Brilliant Numbers
• Pandigital Numbers Formed From the Product of a Number and its Reversal
• Golden Semiprimes
• An Unhappy Family
• A Prime to Remember
• 2019: A Numerical Profile
• Semiprime Factor Ratios
• More about Golden Semiprimes

I was prompted to make yet another post about them because today I turned 25661 days old and was little of interest to found about this number in either the OEIS, Numbers Aplenty or any other sources. However, it is a semiprime, being a product of 67 and 383. I thought I'd examine the number is a more detailed, two dimensional way. Figure 1 illustrates my approach.

Figure 1

I've revisited some old territory in the notes contained in Figure 1 which I've reproduced below:

The number 25661 is a semiprime because it has prime factors of 67 and 383. It can be visualised in two dimensions as a rectangle with a width of 67 units and length of 383 units. As such, its area of course is 25661 square units and its perimeter is 900 units. The ratio of the rectangle's width to its length is thus 67 383 or approximately 0.17493 and the ratio of length to width is 383: 67 or approximately 5.7164. The length of the diagonal of this rectangle is approximately equal to 388.8. The area of the rectangle is equivalent to the sum of the areas of 62 different combinations of three squares. An example is shown where the three squares has sides of 86 units, 92 units and 99 units.

By "old territory", I mean the semiprime is envisioned as a rectangle whose width and length are the smaller and larger prime factors whose product is the area of the rectangle. Thus the integers 900 and 25661 are related via a gematria-like connection. The ratio of the sides produce two other related numbers, both rational, and this case:$$ \frac{67}{383} \approx 0.17493 \text{ and its reciprocal } \frac{383}{67} \approx 5.7164$$What's new is that I've considered the length of the rectangle's diagonal which is an irrational number and equal to \( \sqrt {67^2+383^2} = \sqrt {151178} \approx 388.8162 \).

Finally, I've used the fact that 25661 can be expressed a sum of three squares in 62 different ways to represent the rectangle as being equivalent in area to the sum of any of these three squares. In Figure 1, I've used the example of:$$86^2+92^2+99^2=67 \times 383 =25661$$The square numbers correspond to \( 86^2, 92^2 \text{ and } 99^2 \text{ are } 7396, 8464 \text{ and } 9801 \text{ respectively }\). There are another 61 sets of such triplets that can be linked visually with the rectangular representation of 25661. See Figure 2:

Figure 2

Semiprime Factor Ratios

All biprimes (or semiprimes or 2-almost-primes) can be visualised as unique rectangles and all triprimes (or 3-almost-primes) as rectangular prisms. I only intend to deal with biprimes in this post. Let's take a recent biprime, 24581 = 47 x 523, as a starting point. It can be visualised as a rectangle with a width of 47 units and a length of 523 units. It's the ratio of width to length that's of interest. 

A golden semiprime is defined as a number that factors to: 

• \(p \times q\) (with \(p<q\)) and
• \( |p \times \phi-q|<1 \), where \( \phi\) is the golden ratio of \( \dfrac{1+\sqrt 5}{2} \)

Clearly 24581 does not satisfy this condition and not many semiprimes do. The next for me is 27641 which factors to 131 × 211 and where:$$|131\times \phi-211| \approx 0.9624525$$and so it just barely satisfies the criterion. Here is a partial list as shown in OEIS A108540:

6, 15, 77, 187, 589, 851, 1363, 2183, 2747, 7303, 10033, 15229, 16463, 17201, 18511, 27641, 35909, 42869, 45257, 53033, 60409, 83309, 93749, 118969, 124373, 129331, 156433, 201563, 217631, 232327, 237077, 255271, 270349, 283663, 303533, 326423

Presumably there is an infinity of golden semiprimes. There are other ratios of interest, for example pi. Here the number 154 = 7 x 22 could be treated in a manner similar to the golden semiprimes and the question asked as to whether \( |7 \times \pi-22| \) is less than 1. It turns out that it is (0.9911...) and so could perhaps be termed a circular semiprime. The number 15883 = 71 x 223 yields a much closer result (0.053...). Similarly for \(e\), the number 133 = 7 x 19 yields \( |7 \times e - 19| \approx 0.02797 \) and could be termed an Euler semiprime for want of a better term. 

Some semiprimes are not related to special mathematical constants but are nonetheless of interest. For instance, for Friday 26th August 2016 (the day I'm completing this post), my number 24617 = 239 x 103 and the ratio 239:103 can be expressed approximately as 2.32:1 (rounding off 2.320388... to two decimal places). This is very close to the aspect ratio for the current widescreen cinema standard of 2.35:1 or 2.39:1. However, following the pattern for the golden semiprime ratio, the result of \( |103 \times 2.35-239|=3.05 \) and \( |103 \times 2.39-239|=7.17 \) mean that the results are outside the acceptable range (less than 1).

Another way to view the ratio 239:103 is as 0.69883:0.30117 and if we round off to two decimal places, the result is 0.70:0.30 or 70% : 30%. This is the ratio of copper to zinc in so-called Cartridge brass described as follows:

70/30 brass has excellent ductility and good strength. It is often used where its deep drawing qualities are needed. The alloy is the most common brass in sheet form (source).

I guess the concept of the golden semiprime has opened my eyes to other classifications of semiprimes based on other constants such \(e\) and \( \pi\). Expressing the ratio in such a way that both sides sum to 1 is also useful because, as in the case of 0.70:0.30, connections to physical applications can be drawn.

on August 30th 2021