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