It's always interesting to look at the mathematical properties of the number being used to mark the year ahead in the Anno Domini or AD system. At the time of creation of this post, that number is 2022. First and foremost, its factors should be considered and these are 2, 3 and 337 marking it as a so-called sphenic number because it is the product of three distinct primes.
I've written about these sorts of numbers in two posts titled Sphenic Numbers on June 25th 2018 and Sphenic Numbers Revisited on January 1st 2018. All sphenic numbers have exactly eight divisors and in the case of 2022, these are 1, 2, 3, 6, 337, 674, 1011 and 2022.
2022 has the distinction of belonging to OEIS A105936:
A105936 | Numbers that are the product of exactly 3 primes and are of the form prime(\(n\)) + prime(\(n\)+1). |
The initial members are:
8, 12, 18, 30, 42, 52, 68, 78, 138, 172, 186, 222, 258, 268, 410, 434, 508, 548, 618, 668, 762, 772, 786, 892, 906, 946, 978, 1002, 1030, 1132, 1334, 1374, 1446, 1542, 1606, 1758, 1866, 1878, 1948, 2006, 2022, 2252, 2334, 2414, 2452, 2468, 2486, 2572, 2588
It should be noted that not all members of this sequence are sphenic. For example, 12 is a member but it is not a product of three distinct primes because the factor 2 is repeated. In the case of 12, it can be seen that it is the sum of two consecutive primes viz. 5 and 7. For 2022, the two consecutive primes are 1009 and 1013. The fact that they are separated by 4 makes them cousin primes.
Consulting the Online Encyclopaedia of Integer Sequences or OEIS, the second sequence of interest is OEIS A141769:
A141769 | Beginning of a run of 4 consecutive Niven (or Harshad) numbers. |
1, 2, 3, 4, 5, 6, 7, 510, 1014, 2022, 3030, 10307, 12102, 12255, 13110, 60398, 61215, 93040, 100302, 101310, 110175, 122415, 127533, 131052, 131053, 196447, 201102, 202110, 220335, 223167, 245725, 255045, 280824, 306015, 311232, 318800, 325600, 372112, 455422
Harshad or Niven numbers as they are also called are simply numbers that are divisible by their sum of digits. In the case of 2022, it can be seen that it and the three consecutive numbers following it are Harshad. Let's confirm that:$$ \begin{align} \frac{2022}{6}&=337\\ \frac{2023}{7}&=289\\ \frac{2024}{8}&=278\\ \frac{2025}{5}&=405 \end{align}$$ As can be seen such runs are not common. However, it is possible to have runs of up to twenty consecutive Harshad numbers. See Figure 1.
I've written about Harshad numbers in posts titled Harshad Numbers on February 11th 2017 and Harshad Numbers Revisited on June 30th 2018. Figure 1 shows the start of consecutive runs up to 13. Note that the numbers from 1 to 10 are trivially Harshad.
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Figure 1: permalink for calculating runs |
The next interesting property of 2022 is that not only is it a Harshad number but so are all its powers up to the 7th power. Figure 2 confirms this (SOD stands for Sum Of Digits):
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Figure 2: permalink |
This property constitutes OEIS A135192:
A135192 | Numbers \(n\) that raised to the powers from 1 to \(k\) (with \(k \geq 1 \)) are multiple of the sum of their digits (\(n\) raised to \(k\)+1 must not be a multiple). Case \(k\)=7. |
The initial members of the sequence are:
126, 480, 660, 810, 882, 1020, 1134, 1170, 1260, 1320, 1560, 1590, 2022, 3042, 3222, 4662, 4800, 5670, 5940, 6240, 6600, 7110, 7452, 8100, 8442, 8550, 8820, 8880, 9510, 10110, 10200, 10350, 10620, 10890, 11010, 11106, 11130, 11340, 11460, 11700, 11970
A111948 | Admirable Harshad numbers \(n\) such that the subtracted divisor is equal to the digital sum of \(n\). |
24, 42, 114, 222, 402, 2022, 2202, 7588, 8596, 10014, 11202, 12102, 17668, 21102, 27748, 29764, 31002, 32788, 39844, 42868, 43876, 45388, 46396, 48916, 49924, 55972, 56476, 57484, 58492, 65548, 66556, 69076, 70588, 71596, 78148, 81676
A003219 | Self numbers divisible by sum of their digits (or, self numbers which are also Harshad numbers). |
1, 3, 5, 7, 9, 20, 42, 108, 110, 132, 198, 209, 222, 266, 288, 312, 378, 400, 468, 512, 558, 648, 738, 782, 804, 828, 918, 1032, 1098, 1122, 1188, 1212, 1278, 1300, 1368, 1458, 1526, 1548, 1638, 1704, 1728, 1818, 1974, 2007, 2022, 2088, 2112, 2156, 2178
A023523 | a(\(n\)) = prime(\(n\))*prime(\(n\)-1) + 1. |
The initial members of the sequence are with prime(0) being considered as 1:
3, 7, 16, 36, 78, 144, 222, 324, 438, 668, 900, 1148, 1518, 1764, 2022, 2492, 3128, 3600, 4088, 4758, 5184, 5768, 6558, 7388, 8634, 9798, 10404, 11022, 11664, 12318, 14352, 16638, 17948, 19044, 20712, 22500, 23708, 25592, 27222, 28892
4, 6, 18, 42, 66, 144, 282, 384, 408, 450, 522, 564, 618, 672, 720, 732, 744, 828, 858, 1122, 1308, 1374, 1560, 1644, 1698, 1776, 1848, 1920, 2022, 2304, 2412, 2616, 2766, 2778, 2874, 2958, 2970, 3036, 3042, 3240, 3258, 3354, 3360, 3432, 3540, 3594, 3732
In the case of 2022, the two primes are 19603 and 15559 respectively.
This next property of 2022 is quite unusual and took me some time to fully grasp. This property qualifies the number for membership in OEIS A335600:
A335600 | The poor sandwiches sequence. |
2, 1, 110, 10, 1101, 11010, 3, 330, 30, 3303, 33030, 4, 440, 40, 4404, 44040, 5, 550, 50, 5505, 55050, 6, 660, 60, 6606, 66060, 7, 770, 70, 7707, 77070, 8, 880, 80, 8808, 88080, 9, 990, 90, 9909, 99090, 11, 101, 1010, 22, 20, 202, 220, 2022, 2020, 33, 303, 3030, 44, 404, 4040, 55, 505, 5050, 66, 606, 6060, 77
The OEIS comments help explain what it's all about:
Imagine we would have a pair of adjacent integers in the sequence like [1951, 2020]. The sandwich would then be made of the rightmost digit of a(n), the leftmost digit of a(n+1) and, in between, the absolute difference of those two digits. The pair [1951, 2020] would then produce the (poor) sandwich 112.
Why poor? Because a rich sandwich would insert the sum of the digits instead of their absolute difference - that is 132 in this example. Please note that the pair [2020, 1951] would produce the poor and genuine sandwich 011 (we keep the leading zero: these are sandwiches after all, not integers).
Now we want the sequence to be the lexicographically earliest sequence of distinct positive terms such that the successive sandwiches emerging from the sequence rebuild it, digit after digit.
EXAMPLE
The first successive sandwiches are: 211, 101, 011, 011, 101, 033,...
The first one (211) is visible between a(1) = 2 and a(2) = 1; we get the sandwich by inserting the difference 1 between 2 and 1.
The second sandwich (101) is visible between a(2) = 1 and a(3) = 110; we get this sandwich by inserting the difference 0 between 1 and 1.
The third sandwich (011) is visible between a(3) = 110 and a(4) = 10; we get this sandwich by inserting the difference 1 between 0 and 1; etc.
The successive sandwiches rebuild, digit by digit, the starting sequence.
2022 is what is called an untouchable number because it is not equal to the sum of the proper divisors of any number. The untouchable numbers, up to and including 2022, are:
2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, 516, 518, 520, 530, 540, 552, 556, 562, 576, 584, 612, 624, 626, 628, 658, 668, 670, 708, 714, 718, 726, 732, 738, 748, 750, 756, 766, 768, 782, 784, 792, 802, 804, 818, 836, 848, 852, 872, 892, 894, 896, 898, 902, 926, 934, 936, 964, 966, 976, 982, 996, 1002, 1028, 1044, 1046, 1060, 1068, 1074, 1078, 1080, 1102, 1116, 1128, 1134, 1146, 1148, 1150, 1160, 1162, 1168, 1180, 1186, 1192, 1200, 1212, 1222, 1236, 1246, 1248, 1254, 1256, 1258, 1266, 1272, 1288, 1296, 1312, 1314, 1316, 1318, 1326, 1332, 1342, 1346, 1348, 1360, 1380, 1388, 1398, 1404, 1406, 1418, 1420, 1422, 1438, 1476, 1506, 1508, 1510, 1522, 1528, 1538, 1542, 1566, 1578, 1588, 1596, 1632, 1642, 1650, 1680, 1682, 1692, 1716, 1718, 1728, 1732, 1746, 1758, 1766, 1774, 1776, 1806, 1816, 1820, 1822, 1830, 1838, 1840, 1842, 1844, 1852, 1860, 1866, 1884, 1888, 1894, 1896, 1920, 1922, 1944, 1956, 1958, 1960, 1962, 1972, 1986, 1992, 2008, 2010, 2022
These numbers constitute OEIS A005114.
2022 is a primitive abundant number, since it is smaller than the sum of its proper divisors, none of which is abundant.
2022 is a pseudoperfect number, because it is the sum of a subset of its proper divisors which are 1, 2, 3, 6, 337, 674 and 1011. If the subset {337, 674, 1011} is taken then we have 337 + 674 + 1011 = 2020.
2022 is a Zumkeller number, because its divisors can be partitioned in two sets with the same sum (2028). The divisors of 2022 are 1, 2, 3, 6, 337, 674, 1011 and 2022 and these sum to 4056 or 2 x 2028. There are four groupings of two sets satisfying the condition that each sum to 2028. These are:
- 6, 2022 and 1, 2, 3, 337, 674, 1011
- 1, 2, 3, 2022 and 6, 337, 674, 1011
- 6, 337, 674, 1011 and 1, 2, 3, 2022
- 1, 2, 3, 337, 674, 1011 and 6, 2022
There's a lot more that could be said about 2022 but I'll leave off with a reference to "dismal" arithmetic or "lunar" arithmetic as it's apparently been renamed. Here is a link to a PDF file of July 5th 2011 that explains what is meant by dismal arithmetic. It's free to download. The famous N.J.A. Sloane who created the OEIS is a co-author. Here is the abstract:
Dismal arithmetic is just like the arithmetic you learned in school, only simpler: there are no carries, when you add digits you just take the largest, and when you multiply digits you take the smallest. This paper studies basic number theory in this world, including analogues of the primes, number of divisors, sum of divisors, and the partition function.
2022 makes an appearance in lunar arithmetic via OEIS A170806:
A170806 | Primes in lunar arithmetic in base 3 written in base 3. |
In Sloane's paper, there is the following definition:
Theorem 9. In base \(b\) dismal arithmetic, \(n\) is prime if and only if the dismal sum of its distinct dismal prime divisors is equal to \(n\).
I won't go further into this arithmetic in this post but perhaps I will later on.
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