Abundant But Not Zumkeller

A property of the number \( \textbf{28062}\) associated with my diurnal age today prompted me to look more closely at abundant numbers that are not Zumkeller. 28062 has the following properties:$$ \begin{align} 28062 &=2 \times 3^2 \times 1559\\ \text{ divisors }  &\rightarrow 1, 2, 3, 6, 9, 18, 1559, 3118, 4677, 9354, 14031, 28062 \end{align}$$Normally, for abundant numbers, the set of divisors can be divided into two mutually exclusive sets whose elements sum to the same number. However, this simply can't be done with 28062.

Now up to 40000, there are 718 abundant numbers that are abundant but not Zumkeller. 146 of these have an odd sum of divisors and an even split is not possible. The other 572 have an even sum of divisors and thus have the potential for an even split but it proves impossible to find one. Here are the details:

There are 572 abundant numbers with an \( \textbf{even}\) sum of divisors. They are (up to 40000):

[738, 748, 774, 846, 954, 1062, 1098, 1206, 1278, 1314, 1422, 1494, 1602, 1746, 1818, 1854, 1926, 1962, 2034, 2286, 2358, 2466, 2502, 2682, 2718, 2826, 2934, 3006, 3114, 3222, 3258, 3438, 3474, 3492, 3546, 3582, 3636, 3708, 3798, 3852, 3924, 4014, 4068, 4086, 4122, 4194, 4302, 4338, 4518, 4572, 4626, 4716, 4734, 4842, 4878, 4932, 4986, 5004, 5058, 5094, 5274, 5364, 5436, 5526, 5598, 5634, 5652, 5706, 5868, 5958, 6012, 6066, 6228, 6246, 6282, 6354, 6444, 6462, 6516, 6606, 6714, 6822, 6876, 6894, 6948, 7002, 7092, 7146, 7164, 7218, 7362, 7542, 7544, 7578, 7596, 7758, 7794, 7902, 7974, 8028, 8082, 8172, 8226, 8244, 8298, 8334, 8388, 8406, 8604, 8622, 8676, 8766, 8838, 8982, 9036, 9054, 9162, 9252, 9378, 9414, 9468, 9684, 9738, 9756, 9846, 9972, 10026, 10116, 10134, 10184, 10188, 10242, 10278, 10386, 10548, 10566, 10674, 10782, 10818, 10926, 11034, 11052, 11106, 11142, 11196, 11268, 11358, 11412, 11538, 11574, 11646, 11754, 11862, 11898, 11916, 12114, 12132, 12186, 12294, 12438, 12492, 12564, 12618, 12708, 12762, 12924, 12942, 13086, 13194, 13212, 13302, 13374, 13428, 13518, 13626, 13644, 13698, 13788, 13842, 13914, 14004, 14166, 14184, 14292, 14328, 14346, 14436, 14562, 14598, 14724, 14778, 14814, 14886, 14922, 15084, 15102, 15156, 15192, 15354, 15426, 15462, 15516, 15534, 15588, 15786, 15804, 15858, 15894, 15948, 15966, 16056, 16164, 16326, 16344, 16398, 16452, 16488, 16542, 16596, 16668, 16722, 16776, 16812, 16866, 16938, 17046, 17154, 17208, 17244, 17352, 17406, 17478, 17532, 17586, 17676, 17694, 17838, 17946, 17964, 18072, 18108, 18162, 18234, 18324, 18342, 18378, 18504, 18558, 18594, 18702, 18756, 18828, 18882, 18918, 18936, 19098, 19134, 19242, 19368, 19476, 19512, 19566, 19638, 19674, 19692, 19746, 19854, 19944, 19962, 20052, 20106, 20214, 20232, 20268, 20322, 20376, 20484, 20556, 20718, 20754, 20772, 20934, 21078, 21096, 21132, 21258, 21348, 21366, 21474, 21564, 21618, 21636, 21834, 21852, 21906, 22014, 22068, 22104, 22122, 22158, 22212, 22266, 22284, 22300, 22392, 22482, 22536, 22662, 22700, 22716, 22824, 22900, 22986, 23022, 23076, 23094, 23148, 23202, 23238, 23292, 23300, 23346, 23418, 23454, 23508, 23526, 23724, 23742, 23778, 23796, 23832, 23886, 23900, 24100, 24228, 24264, 24372, 24498, 24588, 24606, 24714, 24858, 24876, 24984, 25100, 25128, 25182, 25236, 25362, 25416, 25524, 25614, 25686, 25700, 25722, 25794, 25848, 25884, 25902, 26046, 26118, 26154, 26172, 26262, 26300, 26388, 26424, 26478, 26604, 26658, 26694, 26748, 26766, 26802, 26856, 26874, 26900, 26982, 27036, 27100, 27198, 27252, 27288, 27396, 27414, 27558, 27576, 27684, 27700, 27774, 27828, 27882, 27954, 28008, 28062, 28100, 28206, 28278, 28300, 28332, 28422, 28494, 28584, 28692, 28746, 28818, 28872, 28926, 28962, 29034, 29124, 29142, 29178, 29196, 29286, 29300, 29448, 29466, 29556, 29628, 29772, 29826, 29844, 29934, 30006, 30042, 30168, 30204, 30312, 30474, 30546, 30582, 30700, 30708, 30762, 30852, 30924, 30978, 31014, 31032, 31068, 31100, 31176, 31194, 31300, 31338, 31446, 31554, 31572, 31608, 31662, 31700, 31716, 31788, 31896, 31932, 31986, 32094, 32166, 32202, 32328, 32418, 32598, 32652, 32796, 32814, 32904, 32958, 33084, 33100, 33192, 33246, 33336, 33444, 33498, 33606, 33624, 33678, 33700, 33714, 33732, 33786, 33822, 33876, 34002, 34092, 34218, 34308, 34326, 34434, 34488, 34700, 34758, 34794, 34812, 34900, 34956, 35064, 35082, 35118, 35172, 35300, 35352, 35388, 35514, 35622, 35676, 35766, 35874, 35892, 35900, 35928, 35946, 35982, 36054, 36198, 36216, 36306, 36324, 36468, 36486, 36522, 36648, 36684, 36700, 36702, 36756, 36954, 37116, 37134, 37188, 37242, 37300, 37404, 37458, 37494, 37512, 37566, 37602, 37656, 37764, 37782, 37836, 37900, 37998, 38034, 38196, 38268, 38300, 38322, 38358, 38466, 38484, 38538, 38574, 38754, 38898, 38900, 38952, 39132, 39222, 39276, 39348, 39384, 39492, 39654, 39700, 39708, 39726, 39834, 39924, 39978]

There are 146 abundant numbers with an \( \textbf{odd}\) sum of divisors. These are up to 40000:

[18, 36, 72, 100, 144, 162, 196, 200, 288, 324, 392, 400, 450, 576, 648, 784, 800, 882, 900, 968, 1152, 1296, 1352, 1458, 1568, 1600, 1764, 1800, 1936, 2178, 2304, 2450, 2500, 2592, 2704, 2916, 3042, 3136, 3200, 3528, 3600, 3872, 4050, 4356, 4608, 4624, 4900, 5000, 5184, 5202, 5408, 5776, 5832, 6050, 6084, 6272, 6400, 6498, 7056, 7200, 7744, 7938, 8100, 8450, 8464, 8712, 9216, 9248, 9522, 9604, 9800, 10000, 10368, 10404, 10816, 11025, 11250, 11552, 11664, 12100, 12168, 12544, 12800, 12996, 13122, 13456, 14112, 14400, 15138, 15376, 15488, 15876, 16200, 16900, 16928, 17298, 17424, 18432, 18496, 19044, 19208, 19600, 19602, 20000, 20736, 20808, 21632, 22050, 22500, 23104, 23328, 23716, 24200, 24336, 24642, 25088, 25600, 25992, 26244, 26912, 27378, 28224, 28800, 28900, 30258, 30276, 30752, 30976, 31752, 32400, 33124, 33282, 33800, 33856, 34596, 34848, 36100, 36450, 36864, 36992, 38088, 38416, 39200, 39204, 39762, 40000]

The total number is 718

Note how I've marked 11025 in red. This is because it is the only odd number with an odd number of divisors. We have:$$11025 = 3^2 \times 5^2 \times 7^2 = 105^2 \text{ with 27 divisors}$$The next such number is 99225 where:$$99225=3^4 \times 5^2 \times 7^2 = 315^2 \text{ with 243 divisors}$$

Let's not confuse oddness and evenness of the divisor sums with the oddness and evenness of the numbers themselves. The majority of Zumkeller numbers are even. To quote from an earlier blog of mine:

In the range up to 100,000 there are 24362 even Zumkeller numbers comprising 24.362% of the range. However, there are only 208 odd Zumkeller numbers in that range, comprising 0.208%.

Zumkeller Numbers

 I've been thinking about Zumkeller numbers. Here's a definition:

A number is a Zumkeller number if its divisors can be split into two separate sets that have the same sum. A key requirement for this to be possible is that the total sum of all divisors must be an even number. If the total sum were odd, you couldn't possibly split it into two equal integer sums.

Now the great majority of Zumkeller numbers are \( \textbf{even} \) numbers. In the range up to 100,000 there are 24362, comprising 24.362% of the range. However, there are only 208 \( \textbf{odd} \) Zumkeller numbers in that range, comprising 0.208%. Now how many abundant numbers are \( \textbf{NOT} \) Zumkeller number because their sum of divisors is \( \textbf{odd} \)? In the range up to 100,000, there are 146 such numbers comprising OEIS A156903. Of these, only two are odd, 1025 and 99225 with:

• \(11025 = 3^2 \times 5^2 \times 7^2 = 105^2\)
• \(99225 = 3^4 \times 5^2 \times 7^2\ = 945^2 \)

Additionally there 572 abundant numbers with an even sum of divisors that are \( \textbf{NOT}\) Zumkeller. So overall there are 718 abundant numbers in the range up to 100,000 that aren't Zumkeller. See blog post titled Abundant But Not Zumkeller for the list.

Even 70, the smallest "weird number" (an abundant number that is not the sum of any subset of its proper divisors), is a Zumkeller number. It's sum of divisors is 144 > 140 and thus it is abundant. It's divisors are {1, 2, 5, 7, 10, 14, 35, 70}. These can be split into {70, 2} and {1, 5, 7, 10, 14, 35} with both summing to 72. In fact, up to 100,000 at least, \( \text{all}\) of the weird numbers are Zumkeller numbers. Here are the weird numbers up to 100,000:

70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670, 19810, 20510, 21490, 21770, 21910, 22190, 23170, 23590, 24290, 24430, 24710, 25130, 25690, 26110, 26530, 26810, 27230, 27790, 28070, 28630, 29330, 29470, 30170, 30310, 30730, 31010, 31430, 31990, 32270, 32410, 32690, 33530, 34090, 34370, 34930, 35210, 35630, 36470, 36610, 37870, 38290, 38990, 39410, 39830, 39970, 40390, 41090, 41510, 41930, 42070, 42490, 42910, 43190, 43330, 44170, 44870, 45010, 45290, 45356, 45710, 46130, 46270, 47110, 47390, 47810, 48370, 49070, 49630, 50330, 50890, 51310, 51730, 52010, 52570, 52990, 53270, 53830, 54110, 55090, 55790, 56630, 56770, 57470, 57610, 57890, 58030, 58730, 59710, 59990, 60130, 60410, 61390, 61670, 61810, 62090, 63490, 63770, 64330, 65030, 65590, 65870, 66290, 66710, 67690, 67970, 68390, 68810, 69370, 69790, 70630, 70910, 71330, 71470, 72170, 72310, 72730, 73430, 73570, 73616, 74270, 74410, 74830, 76090, 76370, 76510, 76790, 77210, 77630, 78190, 78610, 79030, 80570, 80710, 81410, 81970, 82670, 83090, 83312, 83510, 84070, 84910, 85190, 85610, 86030, 86170, 86590, 87430, 88130, 89390, 89530, 89810, 90230, 90370, 90790, 91070, 91210, 91388, 91490, 92330, 92470, 92890, 95270, 95690, 96110, 96670, 97930, 98630, 99610, 99890

I've only written about Zumkeller numbers on one previous occasion. That was a post titled Zumkellar Numbers, Half Zumkellar Numbers and Pseudoperfect Numbers that I uploaded in November of 2018. In that post I made reference to the person behind these numbers by quoting a eulogy by Neil Sloane, the originator of the OEIS.

I am deeply sorry to have to report that Reinhard Zumkeller passed away at the end of March 2016. He suffered from pancreatic cancer, which had already progressed to an advanced stage when it was diagnosed. He was a long-time contributor to the OEIS, and was later an editor and then a diligent and dedicated editor-in-chief. Between 2000 and 2016 he contributed over 23000 items to the OEIS. Reinhard was a great Haskell expert, and he was already ready to write a Haskell program and compute 10000 terms when I was studying a new sequence and wanted to see a graph. He will be greatly missed. Neil Sloane, July 3, 2016.

Mathematical Properties of 2022

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.

The initial members of the sequence are:

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.

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):

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

Not only is 2022 a Harshad number but it is also an admirable number, the latter being defined as a number whose sum of proper divisors is equal to the number itself with the proviso that one of the divisors is negative. In the case of 2022, its proper divisors are 1, 2, 3, 6, 337, 674 and 1011 which sum to 2034. However, if the +6 is made -6, then the sum becomes 2022. Moreover, 6 happens to be the digit sum of 2022 since 2 + 2 + 0 + 2 =6. This qualifies 2022 for membership is OEIS A111948: 

A111948

Admirable Harshad numbers \(n\) such that the subtracted divisor is equal to the digital sum of \(n\).

The initial members of the sequence are:

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

2022 is also a self number because there is no number that, when added to its sum of digits, produces 2022. Thus it both a Harshad and a self number which qualifies it for membership in OEIS  A003219:

A003219

Self numbers divisible by sum of their digits (or, self numbers which are also Harshad numbers).

The initial terms of the sequence are:

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 

I've written about self numbers in a post titled Self Numbers and Junction Numbers on October 25th 2018.

The next two interesting properties of 2022 involve primes (as did OEIS A105936 mentioned earlier). The first property qualifies it for admission in OEIS A023523 (permalink):

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

In the case of 2022, it is the product of the 14th prime (43) and the 15th prime (47) plus 1.

The second interesting property of 2022 involving primes qualifies it for membership in OEIS A064403:

A064403

Numbers \(k\) such that prime(\(k\)) + \(k\) and prime(\(k\)) - \(k\) are both primes.

The initial members of this sequence are:

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.

The sequence runs:

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. 

23500: Blast from the Past

One of my earliest posts regarding my diurnal age dates to August 5th 2013 and was uploaded to my Pedagogical Posturing blog. This Mathematical Meandering blog did not exist at that time. This was before I had any notion that OEIS and other resources existed. Rather naively, I wrote in a post titled Reflections on 23500:

Today I'm 23500 days old and I was looking around to see if there was any significance to this number. It's factors are unremarkable (2^2 x 5^3 x 47) but the fact that it's halfway between 23000 and 24000 means that it pops up quite frequently in Internet searches, as would 22500 or 24500 I would imagine. A search reveals that the approximate population of Boston in 1620 was 23500 and there are several towns around the world that are listed as having this population currently e.g. Bishopbriggs in Scotland.

Bishopbriggs grew from a small rural village on the old road from Glasgow to Kirkintilloch and Stirling during the 19th century, eventually growing to incorporate the adjacent villages of Auchinairn, Cadder, Jellyhill and Mavis Valley. It currently has a population of approximately 23,500 people.

Source: http://en.wikipedia.org/wiki/Bishopbriggs

It turns out that Mount Isa has the same population (source):

Mount Isa is located just 200 kilometres from the Northern Territory border and 1,829 kilometres from Queensland’s capital, Brisbane. The nearest major city, Townsville, can be found 883 kilometres from The Isa. Mount Isa covers an area of over 43,310 square kilometres, making it geographically the second largest city in Australia to Kalgoorlie-Boulder, Western Australia ... With a population of approximately 23,500, Mount Isa is a major service centre for north-west Queensland.

Many other examples of towns having populations of about 23500 could be quoted. In addition of populations, the number sometimes comes up as a dollar figure (source):

In its third annual funding cycle, the Black Philanthropy Initiative has pumped $23,500 back into the Winston-Salem area to help African Americans improve their parenting skills.

Interestingly, it turns out that the centre of the Sun is about 23500 times more distant from us than the centre of the Earth (source).

The sun is far enough away (about 23,500 earth radii) that it took a long time before people knew accurately how far away the sun was. Certainly the ancient Greeks had calculated the distance, but they also knew that their results could be off. 

Many countries in the world have five digit postal codes or zip codes as they are sometimes known. These codes identify particular locations within the country e.g. Muang Prachinburi, Prachinburi, Thailand has a postcode of 23500. The United States uses a five digit system but apparently there is no location corresponding to 23500, although there is for 23499 and 23501. 

I've improved the formatting somewhat but that post was made almost eight years ago and is testimony to how little I knew about number properties at the time. With my current skill set, what more can I uncover about 23500? It turns out that it has 18 entries in the OEIS and a couple of them are interesting enough for comment.

The first is its membership of OEIS A133524:

A133524

Sum of squares of four consecutive primes.

The members of the sequence, up to 23500, are:

87, 204, 364, 628, 940, 1348, 2020, 2692, 3700, 4852, 5860, 7108, 8548, 10348, 12220, 14500, 16732, 18580, 21100, 23500

Nowadays I'd write a program in SageMath to generate this sequence. Here is a permalink to that program in SageMathCell.

The second is its membership of OEIS A035959 where \(n=42\):

A035959

Number of partitions of \(n\) in which no parts are multiples of 5.

The members of the sequence, up to 23500, are:

1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, 44, 60, 76, 100, 127, 164, 205, 262, 325, 409, 505, 628, 769, 950, 1156, 1414, 1713, 2081, 2505, 3026, 3625, 4352, 5192, 6200, 7364, 8756, 10357, 12258, 14450, 17034, 20006, 23500

Here is the permalink to the algorithm in SageMathCell that will generate this sequence. 

As we turn to Numbers Aplenty, my other major source of information about numbers, we discover that it is a Harshad number as well as an abundant, practical, pseudoperfect and Zumkeller number. There's lots more information there as well. See Figure 1.

Figure 1: link

Over the years, I've developed programs related to practical, pseudoperfect and Zumkeller numbers. I've stored these as a Google Document accessible via this link.

For example, here is a permalink to a program that will determine all the practical numbers in a given range. Remember a practical number has the property that each smaller number is the sum of distinct divisors of that number. Between 23450 and 23550, there are the following practical numbers:

23450, 23452, 23460, 23472, 23474, 23478, 23484, 23490, 23496, 23500, 23504, 23506, 23520, 23528, 23530, 23532, 23540, 23542, 23544, 23548, 23550

A pseudoperfect number, sometimes called a semiperfect number, is a number in which a subset of its proper divisors sum to the number itself. Nearly all abundant numbers are pseudoperfect and the ones that aren't are called weird. In the case of 23500, there are many such subsets that can be formed from its proper divisors of 1, 2, 4, 5, 10, 20, 25, 47, 50, 94, 100, 125, 188, 235, 250, 470, 500, 940, 1175, 2350, 4700, 5875 and 11750. One such subset is 1175, 4700, 5875 and 11750. Here is permalink to SageMathCell and a program that will display all of the possible subsets.

Zumkeller numbers are related to perfect numbers such as 6, the divisors of which can be written as the set {1, 2, 3, 6}. There are two mutually exclusive subsets of this set, {1, 2, 3}  and {6}, whose union is the original set and both of whose elements add to 6. Zumkeller numbers are similar in that there are two mutually exclusive subsets of this set of divisors, whose union is the original set but with the difference that both of the elements in each subset add to a number other than the originating number. For example, 20 has the set of divisors {1, 2, 4, 5, 10, 20} and there are two subsets {1, 20} and {2, 4, 5, 10} that both total 21. Thus 20 is a Zumkeller number. In the case of 23500, there are many such pairs of subsets. Here is one example:

[4, 10, 94, 250, 2350, 23500] and [1, 2, 5, 20, 25, 47, 50, 100, 125, 188, 235, 470, 500, 940, 1175, 4700, 5875, 11750] which both sum to 26208.

Here is a permalink to a SageMathCell program that will display all the pairs of subsets.

Harshad or Niven numbers are divisible by the sum of their digits. Here is a permalink to a SageMathCell program that displays all such numbers in the range from 23450 to 23550. These numbers are:

23450, 23454, 23457, 23460, 23472, 23490, 23496, 23500, 23505, 23508, 23520, 23526, 23530, 23544, 23550

23500 is formed from a concatenation of the first three prime numbers (2, 3 and 5), followed by two zeroes. It's interesting to note that 23500 written in scientific notation as \(2.35 \times 10^4\) features all the digits from 0 to 5. Earth's axial tilt is close to 23.5° and, as mentioned in my original post, there is this most interesting astronomical detail regarding 23500: 

Distance to Sun = 1.5 x 10^13 cm = 23500 Earth radii = 1 Astronomical Unit (AU) 

So nothing mathematically Earth-shattering in this post but just a reflection on the differences in my mathematical knowledge between now and then.