Reciprocals of Primes

I came across an interesting video on YouTube by Michael Penn in which he investigates an interesting property of prime numbers, namely that if $p$ is an odd prime, then:$$\frac{2}{p}=\frac{1}{m}+\frac{1}{n}\text{ where }m>n\ge 1\text{ and }m,n\in \mathbb{Z}$$Furthermore, this representation is unique. The proof is simple enough and yet quite elegant. Here it is:$$\begin{array}{rl}\frac{2}{p}& =\frac{1}{m}+\frac{1}{n}\\ 2mn& =np+mp\\ 2mn-np-mp& =0\\ 4mn-2np-2mp& =0\\ 4mn-2np-2mp+p^2& =0+p^2\\ (2m-p)(2n-p)& =p^2\end{array}$$This is the crucial point. $p^2$ factorises to either $p\times p$ or $p^2\times 1$. However, because $m>n$, then $2m-p>2n-p$ and so we need to look at the factors of $p^2$ and $1$. So, because $p^2>1$, we have:$$\begin{gathered} 2m-p=p^2\text{ and }2n-p=1 \\ m=\frac{p(p+1)}{2}\text{ and }n=\frac{p+1}{2} \end{gathered}$$The $p\times p$ factorisation leads to the trivial case, where $m=n=p$. Figure 1 shows the results for the odd primes up to 97.

Figure 1

Thus we can write ${\displaystyle \frac{2}{97}}={\displaystyle \frac{1}{4753}}+{\displaystyle \frac{1}{49}}$. If a composite number is entered into the above calculations, there will be more than one representation. For example, in the case of 25, we can write:$$\frac{2}{25}=\frac{1}{75}+\frac{1}{15}=\frac{1}{325}+\frac{1}{13}$$The reason is that the possible factorisations are no longer just $p\times p$ and $p^2\times 1$. In the case of 25, we now have 625 x 1 as well as 125 x 5 and of course the trivial 25 x 25. Hence the two solutions shown above. Here is the embedded video:

It was only when reading some of the comments to this video that I came across a connection with the harmonic mean. For two numbers, $m$ and $n$, the harmonic mean is defined to be $\frac{2mn}{m+n}$. If we go back to our original expression, we see that:$$\begin{array}{rl}\frac{2}{p}& =\frac{1}{m}+\frac{1}{n}\\ \frac{2}{p}& =\frac{m+n}{mn}\\ \frac{p}{2}& =\frac{mn}{m+n}\\ p& =\frac{2mn}{m+n}\end{array}$$So $p$ represents the harmonic mean of the numbers $m$ and $n$. For example, the harmonic mean of 49 and 4753 is 97 (referring back to Figure 1). The harmonic mean is one of the Pythagorean means, the other two being the arithmetic mean and the geometric mean. I did mention these means briefly in a post from 13th September 2020 titled Root-Mean-Square and Other Means but I should investigate this topic in more detail.

An Interesting Iteration

Having turned 26506 days old today, my attention was drawn to this OEIS sequence:

A219960

Numbers which do not reach zero under the repeated iteration $x\to \lceil \sqrt{x}\rceil \times (\lceil \sqrt{x}{\rceil }^2-x)$.

Figure 1 shows that 26506 is the 511th such number and thus the frequency of such numbers is about 1.93%.

Figure 1

The first members of this sequence are as follows:

366, 680, 691, 1026, 1136, 1298, 1323, 1417, 1464, 1583, 1604, 1702, 2079, 2125, 2222, 2223, 2374, 2507, 2604, 2627, 2821, 2844, 2897, 3152, 3157, 3159, 3183, 3210, 3231, 3459, 3697, 3715, 3762, 3802, 3866, 3888, 3936, 3948, 4004, 4111, 4133, 4145, 4231, 4299, ...

Here is a permalink to the algorithm on SageMathCell that will return all members of OEIS A219960 up to and including 26506. 

The OEIS comments include the following conjectures:

• Conjecture 1: All numbers under the iteration reach 0 or, like the elements of this sequence, reach a finite loop, and none expand indefinitely to infinity. 

• Conjecture 2: There are an infinite number of such finite loops, though there is often significant distance between them. 

• Conjecture 3: There are an infinite number of pairs of consecutive integers in this sequence despite being less abundant than in A219303.

OEIS A219303 refers to the iterative process where the ceiling function is replaced by the floor function. So what happens to 26506 under this iteration? Here is the trajectory:

26506, 10269, 13770, 18172, 7155, 5950, 10452, 16171, 27264, 48472, 81549, 70642, 30324, 52675, 51750, 53352, 2079, 1702, 2604, 5200, 9417, 18326, 23120, 44217, 64144, 94488, 115808, 161293, 125022, 104076, 81719, 22022, 26671, 36900, 67357, 63180, 81648, 42328, 22248, 37800, 43875, 47250, 59732, 71785, 10452

As can be seen, after five steps a loop of length 38 is entered with a length of 43 steps overall. Figure 2 shows this trajectory using a log scale for the vertical axis.

Figure 2

Up to 26506, the trajectory of maximum length is associated with the number 25923 that has a trajectory of length 86 and ends in 0:

25923, 52002, 100531, 188574, 283185, 481832, 829135, 716046, 1154461, 1251300, 963459, 849430, 602988, 575757, 245916, 49600, 28767, 22610, 28841, 10030, 17271, 20196, 36179, 57682, 96159, 174782, 326401, 447876, 686080, 962469, 1821610, 1201500, 2094173, 3664888, 4475355, 4445716, 4565985, 1675408, 2094015, 3893672, 5929896, 10231200, 7680799, 8828820, 11781008, 15383273, 26111488, 3127320, 3610529, 6220072, 12357735, 15895836, 1327671, 2003914, 1617072, 1160064, 2177560, 1499616, 1236025, 577128, 358720, 48519, 71162, 33909, 58460, 25168, 17967, 34830, 25993, 40662, 28684, 36720, 27648, 40247, 30954, 3872, 6111, 10270, 13668, 2457, 2150, 2773, 1908, 1232, 2304, 0

Figure 3 shows the trajectory of 25923 using a log scale for the vertical axis.

Figure 3

Figure 4 shows the distribution of trajectory lengths between 1 and 26506. All square numbers immediately become zero under the iteration. Using 25 as an example, we get:

$\lceil \sqrt{25}\rceil \times (\lceil \sqrt{25}{\rceil }^2-25)=5\times (25-25)=5\times 0=0$

Figure 4

So far only the numbers up to and including 26506 have been examined because the algorithm is processor intensive. However, if we search from 26507 to 50000, we find that the record length increases slightly to 91, again ending in 0. Here is the record length attained by 35727:

35727, 70870, 111873, 117920, 143104, 203523, 353012, 602735, 772338, 266337, 492184, 435240, 237600, 265472, 404544, 780325, 999804, 196000, 110307, 193806, 297675, 240786, 144845, 120396, 4511, 7684, 5280, 3577, 1380, 2432, 3400, 4779, 8470, 16647, 32890, 42588, 54027, 61046, 113584, 223080, 306977, 581640, 403627, 552684, 633888, 1052837, 1943084, 211888, 291813, 469588, 691488, 612352, 577071, 402040, 752475, 823732, 664656, 979200, 891000, 128384, 178423, 214038, 153253, 161112, 197784, 107245, 111192, 121576, 78525, 122516, 240435, 317186, 513240, 608733, 959068, 1305360, 1244727, 813564, 36080, 3800, 2728, 4293, 4158, 4355, 66, 135, 108, 143, 12, 16, 0

Here is the permalink for this calculation. Note that the penultimate number in the trajectory is 16 which is a square number ($4^2$), just as the penultimate number for the previous record trajectory was 2304, also a square number (${48}^2$). Clearly, it is only when a square number is reached in the trajectory that a result of zero will arise in the next iteration. However, in the case of over 98% of numbers (at least in the range up to 26506), the trajectory does not terminate at zero but instead enters a loop.

Conjecture 3, included earlier, states that "there are an infinite number of pairs of consecutive integers" so let's investigate this further. In the range up to 26506, the following pairs occur:

• 2222 2223 
• 8399 8400 
• 11457 11458 
• 12950 12951 
• 19005 19006 
• 19847 19848 
• 22444 22445 
• 23597 23598 
• 25089 25090 
• 25175 25176 
• 25742 25743 

So eleven pairs in that range shows that pairs of such numbers are not that common and of course there's no way to confirm that there are an infinite number of them.

Fee, Phi, Fo, Sum

Time to return to integrals for a while and practice my LaTeX skills. I came across an interesting video on YouTube recently that investigated the following integral:$${\int }_0^{\mathrm{\infty }}\frac{1}{(1+x^{\varphi }{)}^{\varphi }}\text{d}x$$Figure 1 shows that the result is 1 using GeoGebra, using 1000 as the limit of integration rather than infinity, because the program doesn't seem to cope with the latter. 

Figure 1

The program however, gives no clue as to how this result was arrived at, although it looks to be likely given the appearance of the area under the curve. Symbolab isn't much help. See Figure 2.

Figure 2

The online integral calculator wasn't any help either. See Figures 3 and 4.

Figure 3

Figure 4

So to show why the integral is equal to 1, I'll basically follow the steps as outlined in the video. Let's remember that $\varphi$ is the solution to the equation:$$\begin{array}{rl}{x}^2-x-1& =0\\ \text{where }x& =\frac{1+\sqrt{5}}{2}=\varphi \\ \text{also }1& ={\varphi }^2-\varphi \\ \text{and }\frac{1}{\varphi }& =\varphi -1\end{array}$$To integrate, the following substitution is used:$$\begin{array}{rl}u& =x^{\varphi }\\ \text{d}u& =\varphi x^{\varphi -1}\text{d}x\\ \frac{\text{d}u}{\varphi x^{\varphi -1}}& =\text{d}x\end{array}$$The limits of integration don't need to be changed because $u=0$ when $x=0$ and $u\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$. So the integral becomes:$${\int }_0^{\mathrm{\infty }}\frac{\text{d}u}{(1+u{)}^{\varphi }\varphi x^{\varphi -1}}$$ However ${\displaystyle \frac{u}{x}}=x^{\varphi -1}$ because $u=x^{\varphi }$ and so the integral becomes:$$\frac{1}{\varphi }{\int }_0^{\mathrm{\infty }}\frac{x}{(1+u{)}^{\varphi }u}\text{d}u$$Because $u=x^{\varphi }$, we can write $u^{1/\varphi }=x$, thus the integral now becomes:$$\frac{1}{\varphi }{\int }_0^{\mathrm{\infty }}\frac{u^{1/\varphi }}{(1+u{)}^{\varphi }u}\text{d}u$$Now we saw earlier that ${\displaystyle \frac{1}{\varphi }}=\varphi -1$ and we can use this fact in transforming the integral even further. We can now write it as:$$\begin{array}{rl}\frac{1}{\varphi }{\int }_0^{\mathrm{\infty }}\frac{u^{\varphi -1}}{(1+u{)}^{\varphi }u}\text{d}u& =\frac{1}{\varphi }{\int }_0^{\mathrm{\infty }}\frac{u^{\varphi -1-1}}{(1+u{)}^{\varphi }}\text{d}u\\ & =\frac{1}{\varphi }{\int }_0^{\mathrm{\infty }}\frac{u^{\varphi -1-1}}{(1+u{)}^{\varphi -1+1}}\text{d}u\end{array}$$Now this last transformation of the integral may seem strange but it's usefulness becomes apparent once we bear in mind the beta function, defined as:$$\beta (x,y)={\int }_0^{\mathrm{\infty }}\frac{u^{x-1}}{(1+u{)}^{x+y}}\text{d}u=\frac{\mathrm{\Gamma }(x)\mathrm{\Gamma }(y)}{\mathrm{\Gamma }(x+y)}$$In this beta function, if we let $x=\varphi -1$ and $y=1$, our integral now becomes:$$\begin{array}{rl}\frac{1}{\varphi }\beta (\varphi -1,1)& =\frac{1}{\varphi }\frac{\mathrm{\Gamma }(\varphi -1)\mathrm{\Gamma }(1)}{\mathrm{\Gamma }(\varphi )}\\ & =\frac{1}{\varphi }\frac{(\varphi -2)!0!}{(\varphi -1)!}\\ & =\frac{1}{\varphi }\frac{1}{\varphi -1}\\ & =\frac{1}{{\varphi }^2-\varphi }\\ & =1\end{array}$$Thus we have confirmed that the integral does indeed evaluate to 1. The transformation of the gamma function to the factorial is achieved via the fact that $\mathrm{\Gamma }(x)=(x-1)!$.

Here is the actual video embedded into this blog. It covers precisely the same steps and my main purpose in creating this blog is simply to prevent my LaTeX skills from becoming too rusty.

Counting People with Mid-Millennium Numbers

Yesterday marked a milestone of sorts. I reached a diurnal age of 26500, a somewhat unremarkable number commemorated with the tweet shown in Figure 1.

Figure 1

I appended two screenshots to the tweet, one from the OEIS (Figure 2) and the other from Numbers Aplenty (Figure 3):

Figure 2

Figure 3

For some reason, I was prompted to think about what the representation of 26500 might be in Roman numerals. It turns out to be as shown in Figure 4.

Figure 4

So what has any of this to do with counting people? Well, nothing really but as I was investigating the number's properties, it occurred to me that this would be a popular rounding number for populations of towns and islands or communities with common interests or characteristics. This can be confirmed from the following data:

• Tick-borne encephalitis, TBE, has been observed in Aland Islands (population 26,500) for more than 60 years (source). Interestingly:

The self-governing province of the Åland Islands lies off the southwest coast of Finland. Åland is an autonomous, demilitarised, Swedish-speaking region of Finland. Åland consists of more than 6,700 islands, but the current population of over 30,000 live on only 60 islands. Source.

• The most recent mid-year population estimates (2020) for the Outer Hebrides gives a population of 26,500. This shows a decrease of 0.8% (220 persons) from mid 2019 to mid 2020. Source. 
  
  
• Unemployment in the first quarter of 2020: Further rise in Gaza’s unemployment rate; 26,500 people lost their jobs even before the pandemic. Source.
  
  
• CDC estimates Salmonella bacteria cause about 1.35 million infections, 26,500 hospitalisations, and 420 deaths in the United States every year. Food is the source for most of these illnesses. Source.
  
  
• On 30 June 2020, the population of Na h-Eileanan Siar was 26,500. This is a decrease of 0.8% from 26,720 in 2019. Over the same period, the population of Scotland increased by 0.0%. Source.

• The Aromanians in Romania are a non-recognised ethnic minority in Romania that numbered around 26,500 people in 2006. Source.
  
  
• At June 2011, there were 1.01 million children under 15 years of age in Victoria, with boys outnumbering girls by 26,500. This age group comprised 18% of the total Victorian population, down from 20% in June 2001. Source.
  
  
• Published estimates of the risk of vaccine-induced thrombotic thrombocytopenia (VITT) from countries with moderate to high data quality range from 1 case per 26,500 to 1 case per 127,300 first doses of AstraZeneca/COVISHIELD administered. Source.
  
  
• The number of dwellings in Camden Council is forecast to grow from 26,500 in 2016 to 79,631 in 2036, with the average household size falling from 3.08 to 3.01 by 2036. Source.
  
  
• Welcome to the official website of Nicholas County -  a safe, rural community located in the heart of central WV.  County Seat - Summersville; Population: 26,500; Home of WV's largest lake  in the county. Source.
  
  
• February 11, 2021 — The number of vaccine doses available for New York City each week received a 26,500-dose boost through a federal program that provides retail pharmacies with the COVID-19 vaccine. Source.
  
  

The list goes on indefinitely. Of course, the same would hold true of other mid-millennium numbers like 25500 and 27500. For example:

The increase in the number of students inside Midland ISD compared to the start of the previous school year will top 1,000. Superintendent Orlando Riddick said Tuesday the district’s student population appears to not only have eclipsed 25,000 but will end up closer to 25,500. September 2017. Source.

So I've coined this special term, mid-millennium numbers, to describe a class of numbers that are commonly used when quantifying data that cannot be precisely determined. These are to be preferred to millennium numbers such as 26,000 which appear a little too "approximate" (having only two significant figures as opposed to the three of 26,500).

For the next mid-millennium number (27500) see Another Mid-Millennium Number.

SUPER SUPER-PRIME NUMBERS

Super-prime numbers (also known as higher-order primes or prime-indexed primes or PIPs) are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. The subsequence begins:

3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991, ... (OEIS A006450).

The number associated with my diurnal age today, 26489, is one such super-prime because it is the 421st prime and the number 421 is prime. So what is a super super-prime? Well, this is a terminology of my own invention, but I think it aptly describes numbers such as 26489 that have the following properties:

A331031

The prime numbers that are prime-indexed primes and whose digit sum, adjacent digit sum concatenation, and adjacent digit difference concatenation are also primes.

It took me a while to take that all in so let's take it step by step for the case of 26489:

• prime-indexed prime: we have seen that 26489 is the 421st prime and that 421 is prime
  
  
• digit sum is prime: the digits of 26489 total 29 and 29 is prime
  
  
• adjacent digit sum concatenation is prime: 
• 2+6 =8, 6+4=10, 4+8=12, 8+9=17
• these adjacent digit sums give 8101217 when concatenated
• 8101217 is prime
  
  
• adjacent digit difference concatenation is prime:
• |2-6|=4, |6-4|=2, |4-8|=4, |8-9|=1
• these adjacent digit difference give 4241 when concatenated
• 4241 is prime

Here is a permalink to SageMathCell if you want to test out the algorithm for generating the sequence of "super super-primes". Up to 20425103, there are 267 members. These are shown below:

41, 83, 401, 2063, 6863, 10909, 20063, 26489, 44621, 105229, 187067, 205507, 233267, 238547, 240047, 243301, 256307, 346763, 367021, 376003, 395581, 555707, 562181, 563467, 600203, 613243, 644843, 675263, 689789, 785801, 787601, 837667, 845381, 954263, 959389, 1070203, 1089463, 1379029, 1394389, 1550503, 1759489, 1777609, 1868567, 1948603, 1994143, 2002001, 2003321, 2034521, 2071481, 2104547, 2106389, 2184101, 2191529, 2217443, 2231407, 2298389, 2303681, 2312621, 2316203, 2334281, 2342309, 2362163, 2365201, 2387003, 2395747, 2416163, 2458747, 2473067, 2491007, 2501243, 2502767, 2505263, 2578403, 2610701, 2612521, 2629307, 2717129, 2742521, 2775781, 2824447, 2858747, 2877221, 2940521, 2960381, 3030409, 3058201, 3080729, 3161309, 3267067, 3339607, 3429667, 3489007, 3510509, 3528409, 3551881, 3598981, 3623401, 3643403, 3765589, 3775043, 3895981, 4143401, 4169621, 4277263, 4349089, 4364501, 4466443, 4576601, 4615601, 4645181, 4664263, 4928389, 4950409, 4979563, 5010407, 5043881, 5048921, 5049203, 5071103, 5110103, 5115203, 5135621, 5165707, 5297909, 5374307, 5533043, 5533681, 5535281, 5969309, 6000809, 6068443, 6146303, 6246029, 6260629, 6310243, 6345067, 6348781, 6405989, 6525643, 6535163, 6678109, 6740743, 6747421, 6856589, 7014881, 7161103, 7410889, 7415743, 7708009, 7813301, 8030963, 8108921, 8152447, 8207363, 8261381, 8512267, 8618567, 8669981, 8715181, 8720947, 8753707, 8787089, 8846429, 8854981, 8884621, 9061447, 9077521, 9277381, 9297907, 9302467, 9476647, 9792301, 9802343, 9913081, 9972343, 9974509, 9998701, 10013747, 10015903, 10045421, 10067809, 10070201, 10121143, 10180481, 10205207, 10223267, 10246729, 10330367, 10490863, 10500229, 10847621, 10893767, 10990121, 11113547, 11203301, 11228207, 11245547, 11265707, 11310647, 11608489, 11638903, 11725001, 11731963, 11878967, 12004309, 12054403, 12079121, 12382663, 12523909, 12579647, 12867409, 12987103, 13009303, 13162909, 13204847, 13248409, 13474789, 13609963, 13702301, 13836101, 13853263, 13918601, 14306203, 14400707, 14412407, 14504267, 14520403, 14637101, 14833543, 14918509, 15168529, 15230321, 15338801, 15439429, 15471889, 15616967, 15650321, 15944389, 16206103, 16420189, 16509343, 16578103, 16970143, 17046889, 17059843, 17309863, 17533189, 17653243, 18053389, 18223003, 18319243, 18329329, 18486581, 18505181, 18590563, 18665629, 18805667, 18970547, 19721201, 19948363, 19999303, 20132401, 20170421, 20273243, 20314867, 20390221, 20425103

Catch-22 Numbers

On October 4th of 2021, I turned 26482 days old and noted that the sum of its digits is 22. Now 22 is considered a very powerful number in numerology, along with 11 and 33. Not only do the digits of 26482 add to 22 but the number is "framed" by 22:

26482

Digit sum is 22

If the digits of a number contain the digit 2 exactly twice and if the sum of the digits is a multiple of 22, then we might term such numbers Catch-22 numbers. How many of them are there, up to the one million mark? Well, it turns out not that many. There are 6360 such numbers representing 0.636 percent of the total. The minimum is 2299 and the maximum is 992200. Here is a permalink to the SageMathCell calculation.

Catch-22 numbers

First member is 2299

Digit sum is 22

What if the number itself was a multiple of 22? Applying this criterion leads not surprisingly to a significant reduction in the numbers. There are 522 such numbers representing 0.0522 percent of all the numbers up to one million. The minimum is now 2992 and the maximum remains the same (992200). Such numbers might be termed Super Catch-22 numbers. Here is a permalink to the SageMathCell calculation.

First member is 2992

Digit sum is 22

2992 = 22 x 136

Here are the 522 Super Catch-22 numbers in the range up to one million:

[2992, 9922, 12298, 12892, 19228, 19822, 22198, 22396, 22594, 22990, 23782, 24772, 25762, 26752, 27742, 28732, 29128, 29326, 29524, 29920, 32296, 32692, 39226, 39622, 42592, 49522, 52294, 52492, 59224, 59422, 62392, 69322, 73282, 74272, 75262, 76252, 77242, 78232, 82192, 89122, 92092, 92290, 99022, 99220, 102982, 108922, 112288, 112882, 118228, 118822, 121792, 122188, 122386, 122584, 122980, 123772, 124762, 125752, 126742, 127732, 128128, 128326, 128524, 128920, 129712, 132286, 132682, 138226, 138622, 142582, 148522, 152284, 152482, 158224, 158422, 162382, 168322, 171292, 173272, 174262, 175252, 176242, 177232, 179212, 182182, 188122, 192082, 192280, 198022, 198220, 200992, 201982, 202378, 202576, 202774, 203962, 204952, 205942, 206932, 207328, 207526, 207724, 208912, 209902, 210298, 210892, 211288, 211882, 212476, 212674, 213268, 213862, 214258, 214852, 215248, 215842, 216238, 216832, 217426, 217624, 218218, 218812, 219208, 219802, 220198, 220396, 220594, 220990, 221188, 221386, 221584, 221980, 223168, 223366, 223564, 223960, 224158, 224356, 224554, 224950, 225148, 225346, 225544, 225940, 226138, 226336, 226534, 226930, 228118, 228316, 228514, 228910, 229108, 229306, 229504, 229900, 230296, 230692, 231286, 231682, 232078, 232474, 232870, 233266, 233662, 234256, 234652, 235246, 235642, 236236, 236632, 237028, 237424, 237820, 238216, 238612, 239206, 239602, 240592, 241582, 242176, 242374, 242770, 243562, 244552, 245542, 246532, 247126, 247324, 247720, 248512, 249502, 250294, 250492, 251284, 251482, 252076, 252670, 253264, 253462, 254254, 254452, 255244, 255442, 256234, 256432, 257026, 257620, 258214, 258412, 259204, 259402, 260392, 261382, 262174, 262570, 263362, 264352, 265342, 266332, 267124, 267520, 268312, 269302, 272074, 272470, 277024, 277420, 280192, 281182, 282370, 283162, 284152, 285142, 286132, 287320, 288112, 289102, 290092, 290290, 291082, 291280, 293062, 293260, 294052, 294250, 295042, 295240, 296032, 296230, 298012, 298210, 299002, 299200, 302962, 306922, 312268, 312862, 316228, 316822, 320782, 321772, 322168, 322366, 322564, 322960, 323752, 324742, 325732, 326128, 326326, 326524, 326920, 327712, 328702, 332266, 332662, 336226, 336622, 342562, 346522, 352264, 352462, 356224, 356422, 362362, 366322, 370282, 371272, 373252, 374242, 375232, 377212, 378202, 382162, 386122, 392062, 392260, 396022, 396220, 402952, 405922, 412258, 412852, 415228, 415822, 420772, 421762, 422158, 422356, 422554, 422950, 423742, 424732, 425128, 425326, 425524, 425920, 426712, 427702, 432256, 432652, 435226, 435622, 442552, 445522, 452254, 452452, 455224, 455422, 462352, 465322, 470272, 471262, 473242, 474232, 476212, 477202, 482152, 485122, 492052, 492250, 495022, 495220, 502942, 504922, 512248, 512842, 514228, 514822, 520762, 521752, 522148, 522346, 522544, 522940, 523732, 524128, 524326, 524524, 524920, 525712, 526702, 532246, 532642, 534226, 534622, 542542, 544522, 552244, 552442, 554224, 554422, 562342, 564322, 570262, 571252, 573232, 575212, 576202, 582142, 584122, 592042, 592240, 594022, 594220, 602932, 603922, 612238, 612832, 613228, 613822, 620752, 621742, 622138, 622336, 622534, 622930, 623128, 623326, 623524, 623920, 624712, 625702, 632236, 632632, 633226, 633622, 642532, 643522, 652234, 652432, 653224, 653422, 662332, 663322, 670252, 671242, 674212, 675202, 682132, 683122, 692032, 692230, 693022, 693220, 702328, 702526, 702724, 712426, 712624, 720742, 721732, 723712, 724702, 732028, 732424, 732820, 742126, 742324, 742720, 752026, 752620, 762124, 762520, 770242, 771232, 772024, 772420, 773212, 774202, 782320, 801922, 802912, 811228, 811822, 812218, 812812, 820732, 821128, 821326, 821524, 821920, 822118, 822316, 822514, 822910, 823702, 831226, 831622, 832216, 832612, 841522, 842512, 851224, 851422, 852214, 852412, 861322, 862312, 870232, 873202, 881122, 882112, 891022, 891220, 892012, 892210, 900922, 902902, 910228, 910822, 912208, 912802, 920128, 920326, 920524, 920920, 921712, 922108, 922306, 922504, 922900, 930226, 930622, 932206, 932602, 940522, 942502, 950224, 950422, 952204, 952402, 960322, 962302, 971212, 980122, 982102, 990022, 990220, 992002, 992200]