2019: A Numerical Profile

With the coming year, 2019, less than two months away, I decided to investigate some of the numerical properties adhering to this number. Right off the bat, we can see that its digit sum is 12 and, because 3 divides 12, we know that 3 will divide 2019 as well. In fact, 2019 has prime factors of 3 and 673. Thus

2019 = 3 * 673

This means that 673 AD and 1346 AD (673 + 673 = 1346) could be associated with 2019 AD. 673 AD was a time of great expansion for the Islamic world (Muhammad had died in 632 AD). The following year, the Siege of Constantinople (one of many over the centuries) began, but "in 672–673 Arab fleets secured bases along the coasts of Asia Minor, and then proceeded to install a loose blockade around Constantinople." Here are some more details :

The First Arab Siege of Constantinople in 674–678 was a major conflict of the Arab–Byzantine wars, and the first culmination of the Umayyad Caliphate's expansionist strategy towards the Byzantine Empire, led by Caliph Mu'awiya I. Mu'awiya, who had emerged in 661 as the ruler of the Muslim Arab empire following a civil war, renewed aggressive warfare against Byzantium after a lapse of some years and hoped to deliver a lethal blow by capturing the Byzantine capital, Constantinople.

As reported by the Byzantine chronicler Theophanes the Confessor, the Arab attack was methodical: in 672–673 Arab fleets secured bases along the coasts of Asia Minor, and then proceeded to install a loose blockade around Constantinople. They used the peninsula of Cyzicus near the city as a base to spend the winter, and returned every spring to launch attacks against the city's fortifications. Finally, the Byzantines, under Emperor Constantine IV, managed to destroy the Arab navy using a new invention, the liquid incendiary substance known as Greek fire. The Byzantines also defeated the Arab land army in Asia Minor, forcing them to lift the siege. The Byzantine victory was of major importance for the survival of the Byzantine state, as the Arab threat receded for a time. A peace treaty was signed soon after, and following the outbreak of another Muslim civil war, the Byzantines even experienced a period of ascendancy over the Caliphate.

1346 AD was also an interesting year. As reported in Wikipedia, it included these events:

• in Spring, a severe Black Death epidemic began its spread at the River Don near the Black Sea, then spread throughout Russia, the Caucasus, and the Genovese provinces within the year
• on April 16th,  the Serbian Empire was proclaimed in Skopje by Dusan Silni, occupying much of South-Eastern Europe
• on July 11th and 12th, Edward III and the English army cross the English Channel, and begin an invasion of France
• on August 26, at the Battle of Crécy, the English defeat the French, in the first European battle where gunpowder is used.

Well, there's not much Mathematics in the history above so best to move on to more mathematical matters. What follows are some interesting facts about 2019 as number.

OEIS A037015: Numbers n with property that, reading binary expansion of n from right to left, run lengths strictly increase. Here we have \( 2019_{10}=11111100011_2 \). The initial members of the sequence are:

0, 1, 3, 6, 7, 14, 15, 28, 30, 31, 57, 60, 62, 63, 120, 121, 124, 126, 127, 241, 248, 249, 252, 254, 255, 483, 496, 497, 504, 505, 508, 510, 511, 966, 993, 995, 1008, 1009, 1016, 1017, 1020, 1022, 1023, 1987, 1990, 2016, 2017, 2019, 2032, 2033, 2040, 2041, 2044

OEIS A158339: Semiprimes that are the sum of four successive semiprimes. Here we have: 501 + 502 + 505 + 511 = 2019 and 501 = 3 * 167, 502 = 2 * 251, 505 = 5 * 101 and 511 = 7 * 73.

The initial members of this sequence are:

39, 94, 106, 118, 146, 158, 185, 201, 221, 254, 302, 365, 427, 473, 485, 519, 537, 589, 633, 655, 707, 723, 749, 767, 842, 851, 869, 901, 1003, 1145, 1205, 1211, 1219, 1247, 1263, 1337, 1349, 1603, 1646, 1681, 1703, 1731, 1797, 1891, 1903, 1937, 2005, 2019

OEIS A193227: Semiprimes p*q such that p+1 and q+1 are semiprimes. 

Here p+1 = 4 = 2 * 2 and q+1 = 674 = 2 * 337 and the initial members of this sequence are:

9, 15, 25, 39, 65, 111, 169, 183, 185, 219, 305, 365, 471, 481, 579, 785, 793, 831, 939, 949, 965, 1191, 1263, 1369, 1371, 1385, 1565, 1623, 1839, 1983, 1985, 2019

OEIS A091431: Happy-go-Lucky numbers: numbers that are both Happy (OEIS A007770) and Lucky (OEIS A000959). Happy numbers are those whose repeated sums  of squares of digits return 1.

The initial members of this sequence are:

1, 7, 13, 31, 49, 79, 129, 133, 193, 219, 319, 331, 367, 391, 409, 487, 655, 673, 739, 931, 937, 1009, 1029, 1039, 1093, 1209, 1233, 1251, 1275, 1281, 1285, 1303, 1309, 1323, 1339, 1533, 1575, 1587, 1599, 1663, 1771, 1857, 1933, 1959, 1995, 2019

OEIS A076408: Sum of first n perfect powers. As Wikipedia defines it: 

In mathematics, a perfect power is a positive integer that can be expressed as an integer power of another positive integer. More formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that \( m^k = n \). In this case, n may be called a perfect k-th power. If k = 2 or k = 3, then n is called a perfect square or perfect cube, respectively. Sometimes 1 is also considered a perfect power (\(1^k = 1\) for any k).

Here n=22 and, as shown in OEIS A001597, the first 22 perfect powers are: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243 with the sequence of progressive sums being: 1, 5, 13, 22, 38, 63, 90, 122, 158, 207, 271, 352, 452, 573, 698, 826, 970, 1139, 1335, 1551, 1776, 2019.

In fact, there are 140 sequences listed in the OEIS that contain a reference to 2019. I've covered some of the more interesting ones, or at least ones that I could understand.

From Numbers Aplenty, we have the following properties:

2019 is the smallest number that can be written in six ways as the sum of the squares of three primes. Here are the prime triplets:

7 11 43, 7 17 41, 11 23 37, 13 13 41, 17 19 37, 23 23 31

Figure 1 is an extract from the Numbers Aplenty page:

FIGURE 1

Semiprime Chains

Today I turned 25399 days old. This number is a semiprime, meaning that it has two prime factors. In the case of 25399, the prime factors are 11 and 2309. However, the number is a member of OEIS A177216: numbers n that are the product of two distinct primes such that \(2n-1, 4n-3, 8n-7, 16n-15, 32n-31, 64n-63 \text{ and } 128n-127\) are also products of two distinct primes. Below is a screenshot of the mathematical tweet to celebrate the occasion.

Evaluating this leads to the following table:

The chain is thus of the form \(2^a \times n- (2^a-1) \) with the integer \(a \geq 0 \).

The initial members of OEIS A177216 are:

11293, 12139, 25399, 31261, 36199, 44869, 49471, 62521, 72397, 83086, 89737, 91705, 98941, 124846, 125041, 134023, 138994, 144793, 164041, 166171, 170431, 173311, 182527, 199543, 224962, 244294, 258169, 259891, 263086, 275281, 277987 

I was interested to see how many of these terms would survive an extension of the process to \(256 \times n - 255 \), \(512 \times n - 511 \), \(1024 \times n - 1023 \) etc.

Extending the process in the same range to \(256 \times n - 255 \) leads to a considerable thinning of the numbers:

31261 36199 44869 49471 62521 72397 83086 138994 173311 182527 224962 259891 277987

Extending the range to \(512 \times n - 511 \) gives:

31261 36199 138994 173311 259891

Extending the range to \(1024 \times n - 1023 \) gives: 173311

So 173311 is the last man standing. Will it stand any further? As it turns out it will. It holds up to \(2048 \times n - 2047 \) but, at \(4096 \times n - 4095 \), there are three prime factors (419, 601 and 2819) and not two. For \(4096 \times n - 4095 \) to hold, we need to extend the range and this leads to 2212801). Below I've listed the semiprime chain associated with the number 173311.

173311 also has the property that is formed by a concatenation of two primes 173 and 311.

The following SAGE code shows that 2212801 is the only number up to 10 million with a prime chain up to \(4096 \times n - 4095 \):

INPUT

for n in range(1,10000000):

    if len(list(divisors(n)))==4:

        if len(list(divisors(2*n-1)))==4:

            if len(list(divisors(4*n-3)))==4:

                if len(list(divisors(8*n-7)))==4:

                    if len(list(divisors(16*n-15)))==4:

                        if len(list(divisors(32*n-31)))==4:

                            if len(list(divisors(64*n-63)))==4:

                                if len(list(divisors(128*n-127)))==4:

                                    if len(list(divisors(256*n-255)))==4:

                                        if len(list(divisors(512*n-511)))==4:

                                            if len(list(divisors(1024*n-1023)))==4:

                                                if len(list(divisors(2048*n-2047)))==4:

                                                    if len(list(divisors(4096*n-4095)))==4:

                                                        print(n),

OUTPUT

2212801

You can try running the above SAGE code using the evaluate button below the SAGE cell. Sometimes it's a little slow but generally it should work. You can modify the code by adding a # in front of a line e.g. # if len(list(divisors(4096*n-4095)))==4: will remove the 4096 condition and apply only the 2048 and lower conditions. Try it out. You may need to refresh the page after each evaluation.

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