P-Composite Numbers

Let's create a class of positive integers called \(p\)-composite integers where \(p\) stands for any prime number. For a number to be so-called, it must satisfy the following criteria:

• it must be composite
• it's sum of digits must equal \(p\)
• \(p\) must be a divisor of the number

Let's take 1679 as an example. It's sum of digits is 23 and 23 divides it to give 73. Thus 1679 can be termed a 23-composite number. The number of 23-composite numbers in the range up to 40000 is 94. The numbers are:

23-composite

1679, 1886, 3749, 3956, 4577, 4784, 4991, 5198, 5819, 6647, 6854, 7268, 7475, 7682, 8096, 8717, 8924, 9338, 9545, 9752, 10787, 10994, 12857, 13478, 13685, 13892, 14099, 14927, 15548, 15755, 15962, 16169, 16376, 16583, 16790, 17618, 17825, 18239, 18446, 18653, 18860, 19067, 19274, 19481, 21758, 21965, 22379, 22586, 22793, 23828, 24449, 24656, 24863, 25277, 25484, 25691, 26519, 26726, 26933, 27347, 27554, 27761, 28175, 28382, 29417, 29624, 29831, 30659, 30866, 31487, 31694, 32729, 32936, 33557, 33764, 33971, 34178, 34385, 34592, 35627, 35834, 36248, 36455, 36662, 37076, 37283, 37490, 37904, 38318, 38525, 38732, 39146, 39353, 39560

Let's work our way up the primes from 2.

2-composite

20, 110, 200, 1010, 1100, 2000, 10010, 10100, 11000, 20000

3-composite

12, 21, 30, 102, 111, 120, 201, 210, 300, 1002, 1011, 1020, 1101, 1110, 1200, 2001, 2010, 2100, 3000, 10002, 10011, 10020, 10101, 10110, 10200, 11001, 11010, 11100, 12000, 20001, 20010, 20100, 21000, 30000

5-composite

50, 140, 230, 320, 410, 500, 1040, 1130, 1220, 1310, 1400, 2030, 2120, 2210, 2300, 3020, 3110, 3200, 4010, 4100, 5000, 10040, 10130, 10220, 10310, 10400, 11030, 11120, 11210, 11300, 12020, 12110, 12200, 13010, 13100, 14000, 20030, 20120, 20210, 20300, 21020, 21110, 21200, 22010, 22100, 23000, 30020, 30110, 30200, 31010, 31100, 32000

7-composite

70, 133, 322, 511, 700, 1015, 1141, 1204, 1330, 2023, 2212, 2401, 3031, 3220, 4102, 5110, 7000, 10024, 10150, 10213, 10402, 11032, 11221, 11410, 12040, 12103, 13111, 13300, 15001, 20041, 20104, 20230, 21112, 21301, 22120, 23002, 24010, 30121, 30310, 31003, 32011, 32200

11-composite

209, 308, 407, 506, 605, 704, 803, 902, 2090, 3080, 4070, 5060, 6050, 7040, 8030, 9020, 10109, 10208, 10307, 10406, 10505, 10604, 10703, 10802, 10901, 20009, 20108, 20207, 20306, 20405, 20504, 20603, 20702, 20801, 20900, 30008, 30107, 30206, 30305, 30404, 30503, 30602, 30701, 30800

13-composite

247, 364, 481, 715, 832, 1066, 1183, 1417, 1534, 1651, 2119, 2236, 2353, 2470, 2704, 2821, 3055, 3172, 3406, 3523, 3640, 4108, 4225, 4342, 4810, 5044, 5161, 5512, 6214, 6331, 7033, 7150, 7501, 8203, 8320, 9022, 10075, 10192, 10309, 10426, 10543, 10660, 11128, 11245, 11362, 11713, 11830, 12064, 12181, 12415, 12532, 13117, 13234, 13351, 13702, 14053, 14170, 14404, 14521, 15106, 15223, 15340, 16042, 16510, 17212, 18031, 19201, 20137, 20254, 20371, 20605, 20722, 21073, 21190, 21307, 21424, 21541, 22009, 22126, 22243, 22360, 22711, 23062, 23413, 23530, 24115, 24232, 24700, 25051, 25402, 26104, 26221, 27040, 28210, 30082, 30316, 30433, 30550, 30901, 31018, 31135, 31252, 31603, 31720, 32071, 32305, 32422, 33007, 33124, 33241, 34060, 34411, 35113, 35230, 36400, 37102

17-composite

476, 629, 782, 935, 1088, 1394, 1547, 1853, 2159, 2465, 2618, 2771, 2924, 3077, 3383, 3536, 3842, 4148, 4454, 4607, 4760, 4913, 5066, 5219, 5372, 5525, 5831, 6137, 6290, 6443, 6902, 7055, 7208, 7361, 7514, 7820, 8126, 8432, 9044, 9350, 9503, 10268, 10574, 10727, 10880, 11186, 11339, 11492, 11645, 11951, 12257, 12563, 12716, 13175, 13328, 13481, 13634, 13940, 14093, 14246, 14552, 14705, 15164, 15317, 15470, 15623, 16082, 16235, 16541, 17153, 17306, 17612, 18071, 18224, 18530, 19142, 19601, 20366, 20519, 20672, 20825, 21284, 21437, 21590, 21743, 22049, 22355, 22508, 22661, 22814, 23273, 23426, 23732, 24038, 24191, 24344, 24650, 24803, 25109, 25262, 25415, 25721, 26027, 26180, 26333, 27251, 27404, 27710, 28016, 28322, 29240, 30158, 30464, 30617, 30770, 30923, 31076, 31229, 31382, 31535, 31841, 32147, 32453, 32606, 32912, 33065, 33218, 33371, 33524, 33830, 34136, 34442, 34901, 35054, 35207, 35360, 35513, 36125, 36431, 37043, 37502, 38114, 38420, 39032

19-composite

874, 1387, 1558, 1729, 2584, 2755, 2926, 3097, 3268, 3439, 3781, 3952, 4294, 4465, 4636, 4807, 5149, 5491, 5662, 5833, 6175, 6346, 6517, 7372, 7543, 7714, 8056, 8227, 8740, 8911, 9082, 9253, 9424, 10279, 10792, 10963, 11476, 11647, 11818, 12673, 12844, 13186, 13357, 13528, 13870, 14383, 14554, 14725, 15067, 15238, 15409, 15580, 15751, 15922, 16093, 16264, 16435, 16606, 17119, 17290, 17461, 17632, 17803, 18145, 18316, 19171, 19342, 19513, 20197, 20368, 20539, 20881, 21394, 21565, 21736, 21907, 22078, 22249, 22591, 22762, 22933, 23275, 23446, 23617, 24472, 24643, 24814, 25156, 25327, 25840, 26182, 26353, 26524, 27037, 27208, 27550, 27721, 28063, 28234, 28405, 29260, 29431, 29602, 30286, 30457, 30628, 30970, 31483, 31654, 31825, 32167, 32338, 32509, 32680, 32851, 33193, 33364, 33535, 33706, 34048, 34219, 34390, 34561, 34732, 34903, 35074, 35245, 35416, 36271, 36442, 36613, 37126, 37810, 38152, 38323, 39007, 39520

29-composite

4988, 7598, 7859, 9686, 9947, 15689, 16994, 17777, 18299, 19865, 22997, 25868, 27695, 27956, 28478, 28739, 29783, 33698, 33959, 35786, 36569, 37874, 38396, 38657, 38918, 39179, 39962

31-composite

8959, 9796, 17887, 25699, 25978, 28489, 28768, 29884, 36859, 37696, 37975, 39649, 39928

37-composite

37999, 38998, 39997

41-composite

37999, 38998, 39997

In the range up to 40000, there are no \(p\)-composite numbers for \(p \gt 41\). I've incorporated identification of these \(p\)-composite numbers into my daily number analysis. One of the properties of these numbers is base-independent: the prime \(p\) will divide the number regardless of the number base used. However, the sum of digits of the number being equal to \(p\) is very much base-dependent.

Forming Palindromes from Factors

 In a blog post titled Why Is 313131 An Interesting Number?, I remarked that:

$$ 313131=3 \times 7 \times 13 \times 31 \times 37$$If we rearrange the order of multiplication we get the following:$$ 313131=7 \times 3 \times 13 \times 31 \times 37$$Concatenating these digits we get the number \(73133137\) which is palindromic.

I went on to look at what other numbers in the range between 28000 and 29000 have this property and came up with the table shown below:

The numbers are thus relatively rare, there being only 27 in a range of 1000 numbers. This represents a density of 2.7%. The numbers are listed below:

28072, 28125, 28194, 28224, 28242, 28273, 28308, 28322, 28332, 28416, 28431, 28448, 28585, 28589, 28593, 28601, 28602, 28609, 28620, 28672, 28685, 28692, 28750, 28800, 28812, 28847, 28951

The factors under consideration here are all PRIME factors. What if we allow factors that are not necessarily prime. Take 28200 as an example:$$ \begin{align} 28200 &= 2 \times 5 \times 3 \times 2 \times 235 \times 2 \\ &\rightarrow 25322352 \end{align}$$The resultant number after concatenation of the factors is palindromic. Notice that the factor 235 is NOT prime.

It turns out that palindromes constructed in this way are relatively frequent. In the range between 28100 and 28300 (a range of only 200), the density is 15.4%. The numbers are:

28104, 28105, 28125, 28126, 28128, 28130, 28140, 28143, 28152, 28160, 28161, 28175, 28179, 28180, 28182, 28188, 28194, 28200, 28224, 28230, 28236, 28242, 28251, 28256, 28266, 28273, 28275, 28280, 28288, 28296, 28300

I've set up my multipurpose algorithm to identify such numbers when they pop up in my diurnal age analysis.

888 Revisited

In November of 2021, I posted about 888 from a mathematical and non-mathematical perspective. Mathematically, I noted that:

• 888 is the smallest cube in which each digit occurs exactly three times. The list up to one million of such numbers is (permalink):

888, 56592, 58524, 65577, 70869, 78183, 496941, 512427, 516267, 517461, 557949, 565920, 581421, 585558, 661959, 711828, 713772, 723627, 724983, 733053, 739563, 764472, 781830, 877242, 988458

• 888 is the only cube in which three digits occur three times. For example, the next number in the previous sequence (56592) has a cube of 181244621426688 but there are four digits that occur three times.
  
  
• 888 the smallest multiple of 24 whose digit sum is 24 and, as well as being divisible by its digit sum, it is divisible by all of its digits.
  
  
• 888 and 24 show up again in the former's membership of OEIS 236661 where 888 counts the number of partitions of 24 that have a standard deviation greater than 2. Permalink.
  
  
• Other properties of 888 include its being a happy, Harshad, Moran, nude, strobogrammatic, modest, congruent, amenable, practical, abundant, pseudoperfect and Zumkeller number (see Numbers Aplenty).
  
  
• The 8's are involved in 888 again thanks to its membership of OEIS A127335.
  
  

A127335

Numbers that are the sum of 8 successive primes.

 The sequence runs:

77, 98, 124, 150, 180, 210, 240, 270, 304, 340, 372, 408, 442, 474, 510, 546, 582, 620, 660, 696, 732, 768, 802, 846, 888

The eight successive primes in the case of 888 are 97, 101, 103, 107, 109, 113, 127 and 131 with an average of 111. Both 111 and 888 are of course repdigits along with the infamous 666 or number of the beast.

• 888 arises in the context of aliquot sequences via OEIS A014360:
  
  

A014360

Aliquot sequence starting at 552.

The sequence begins:

552, 888, 1392, 2328, 3552, 6024, 9096, 13704, 20616, 30984, ...

To quote from Wolfram Alpha:

It has not been proven that all aliquot sequences eventually terminate and become periodic. The smallest number whose fate is not known is 276. There are five such sequences less than 1000, namely 276, 552, 564, 660, and 966, sometimes called the "Lehmer five". 

In this post, I'll revisit 888 but I'll be doing so via 28192, the number associated with my diurnal age today. This number has the interesting property that its square and cube both contain the digit sequence 888:$$ \begin{align} 28192^2 &= 7947\textbf{888}64 \\ 28192^3 &= 22406687653\textbf{888} \end{align} $$There are only four such numbers in the range up to 40000 and they are 3192, 28192, 31920 and 33878. One of these, 31920, is simply a derivative of 3192. Of these four numbers, it is only 28192 that contains three eights when expressed in terms of its prime factors:$$2\textbf{8}192 = 2^5 \times \textbf{88}1$$It could even be written as \(2^2 \times \textbf{8} \times \textbf{88}1 \) so it is very giving in terms of its eightness!

What about other numbers in the range up to 40000 that display three digit repdigits when squared and cubed? Here are the numbers that satisfy:

For repdigit \( \textbf{111}\), there is only one number that satisfies:

• \(10558^2= \textbf{111}471364 \text{ and } 10558^3 =117691466\textbf{111}2\)

For repdigits \( \textbf{222} \) and \( \textbf{333} \), no numbers satisfy but for repdigit \( \textbf{444}\) we have:

• \(6962^2 = 48469\textbf{444} \text{ and } 6962^3 = 337\textbf{444}269128\)
• \(12038^2 = 144913\textbf{444} \text{ and } 12038^3 = 17\textbf{444}68038872\)
• \(21081^2 = \textbf{4444}08561 \text{ and } 21081^3 = 936857687\textbf{444}1\)
• \(32538^2 = 1058721\textbf{444} \text{ and } 32538^3 = 3\textbf{444}8678344872\)
• \(37808^2 = 1429\textbf{444}864 \text{ and } 37808^3 = 540\textbf{444}51418112\)

For repdigit \( \textbf{555} \), again only one number satisfies:

• \(38152^2 =14\textbf{555}75104 \text{ and } 38152^3 = \textbf{555}33101367808\)

For the famous repdigit \( \textbf{666} \), we have two numbers that satisfy with some extra 6's thrown in for the case of \(30605\):

• \(17767^2 = 315\textbf{666}289 \text{ and } 17767^3 = 560844295\textbf{666}3\)
• \(30605^2 = 93\textbf{6666}025 \text{ and }30605^3 = 28\textbf{66666}3695125\)

For the repdigit \( \textbf{777} \), again two numbers satisfy:

• \( 18924^2 = 35811\textbf{777}6 \text{ and }18924^3= 6\textbf{777}020793024 \)
• \( 34753^2 =120\textbf{777}1009 \text{ and } 34753^3 = 41973665875\textbf{777} \)

For the repdigit \( \textbf{999} \) we have a bonanza so I'll just list the numbers and show one example:

\( 9997, 9998, 9999, 19998, 19999, 29999, 38729, 39999 \)

• \(9997^2 = \textbf{999}40009 \text{ and } 9997^3 = \textbf{999}100269973\)

Self-Fibonacci

Here is an interesting sequence generated by a hidden connection to Fibonacci based on the letter-number association shown in Table 1:

Table 1: source

The sequence is OEIS A129938:

A129938: "Self-Fibonacci"; a(n) is the sum of the last nine terms. Sequence starts with 6, 9, 2, 15, 14, 1, 3, 3, 9 which are f, i, b, o, n, a, c, c, i if you consider a=1, b=2, c=3, ..., z=26.

The sequence begins 6, 9, 2, 15, 14, 1, 3, 3, 9, 62, 118, 227, 452, 889, 1764, 3527, 7051, 14099, 28189, ...

I only chanced upon this sequence because 28189, my diurnal age today, is a member. I've written about the various connections between numbers and letters in a post titled Days of the Year and Gematria back in August of 2021. The idea behind this sequence reminds me of my own approach described in a blog post titled Consolidating Fibonacci-like Numbers where I considered numbers whose digits following a Fibonacci pattern e.g. 21347:$$21347 \text{ where }2 + 1 =3, 1+3=4,3+4=7$$However, getting back to approach followed in OEIS A129938, an interesting "spin-off" could be that previously unnamed tribonacci sequences could be given memorable names. For example, using Table 1 we could write:$$ \text{ cat } \rightarrow \text{ c, a, t } \rightarrow \text{ 3, 1, 20 }$$and so the "cat" sequence becomes:$$3, 1, 20, 24, 45, 79, \dots$$Similarly we have:$$ \text{ dog } \rightarrow \text{ d, o, g } \rightarrow \text{ 4, 15, 7 }$$ So the "dog" sequence becomes:$$4, 15, 7, 26, 48, 81, \dots$$Silly I know but it would make for an interesting puzzle in Puzzle of the Day. The sequence doesn't have to be tribonacci, it could simply be Fibonacci-like. For example, we could ask why is the sequence 13, 5, 18, 23, 41, 65, ... egocentric? The answer is that:$$13, 5 \rightarrow \text{ m, e } \rightarrow \text{ me }$$Similarly, the sequence could be made of four or more seeds and a puzzle created. For example, we could ask what does this sequence 13, 9, 12, 11, 45, 77, ... and the Milky Way have in common? The answer is that the first four members of the sequence are the seeds to generate the future members of the sequence and we have:$$ 13, 9, 12, 11 \rightarrow \text{ m, i, l, k } \rightarrow \text{ milk}$$That's enough nonsense for the moment but let's not forget that there has always been a long-standing connection between letters of certain alphabets (Hebrew, Ancient Greek and Arabic for example) and numbers. With the English language the connection has weakened but it's still there and not just in the way shown in Table 1. There are other ways to assign values to letters in the English alphabet. Table 2 shows an alternative way that is more in keeping with the ancient languages.

Table 2: source

Perfect Powers

Firstly a reminder. What is a perfect power?

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 \(k\)).

Today I turned 28188 days old and one of the interesting properties of this number, apart from the digit 8 appearing three times, is that its sum of digits (SOD) and product of digits (POD) are both perfect powers. This is because:$$ \begin{align} \text{SOD}(28188) &= 27 = 3^3\\ \text{POD}(28188) &= 1024 = 2^{10 }\end{align}$$This got me thinking as to how many positive integers in the range up to 40000 have this property. Well, it turns out that there are 217 and here they are (permalink):

4, 8, 9, 18, 22, 44, 81, 88, 144, 224, 242, 333, 414, 422, 441, 448, 484, 844, 999, 1124, 1133, 1142, 1177, 1214, 1224, 1241, 1242, 1313, 1331, 1339, 1393, 1412, 1421, 1422, 1555, 1717, 1771, 1888, 1933, 2114, 2124, 2141, 2142, 2214, 2222, 2241, 2248, 2284, 2411, 2412, 2421, 2428, 2482, 2824, 2842, 3113, 3131, 3139, 3193, 3311, 3319, 3391, 3913, 3931, 4112, 4121, 4122, 4211, 4212, 4221, 4228, 4282, 4444, 4822, 5155, 5515, 5551, 7117, 7171, 7711, 8188, 8224, 8242, 8422, 8818, 8881, 8888, 9133, 9313, 9331, 11114, 11124, 11133, 11141, 11142, 11214, 11222, 11241, 11248, 11284, 11313, 11331, 11411, 11412, 11421, 11428, 11482, 11824, 11842, 12114, 12122, 12141, 12148, 12184, 12212, 12221, 12222, 12411, 12418, 12481, 12814, 12841, 12888, 13113, 13131, 13311, 13399, 13939, 13993, 14111, 14112, 14121, 14128, 14182, 14211, 14218, 14281, 14488, 14812, 14821, 14848, 14884, 18124, 18142, 18214, 18241, 18288, 18412, 18421, 18448, 18484, 18828, 18844, 18882, 19339, 19393, 19933, 21114, 21122, 21141, 21148, 21184, 21212, 21221, 21222, 21411, 21418, 21481, 21814, 21841, 21888, 22112, 22121, 22122, 22211, 22212, 22221, 22228, 22282, 22444, 22822, 24111, 24118, 24181, 24244, 24424, 24442, 24811, 28114, 28141, 28188, 28222, 28411, 28818, 28881, 31113, 31131, 31311, 31399, 31939, 31993, 33111, 33199, 33399, 33919, 33939, 33991, 33993, 39139, 39193, 39319, 39339, 39391, 39393, 39913, 39931, 39933

What about numbers whose sums of divisors are perfect powers? In the range up to 40000, there are only 18 and they are (permalink):

3, 7, 21, 31, 81, 93, 127, 217, 381, 400, 651, 889, 2667, 3937, 8191, 11811, 24573, 27559

Take 21 with divisors of 1, 3, 7 and 21 as an example:$$ \sigma(21)=32=2^5$$What about numbers whose totients are perfect powers? Let's recall that:

The totient of a number, denoted by the Euler's totient function \(\phi(n)\), is the count of positive integers less than or equal to \(n\) that are coprime (relatively prime) to \(n\). Two numbers are coprime if their only common factor is 1.

Numbers of this sort are more frequent with there being 119 in the range up to 40000. They are (permalink):

5, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285, 1360, 1536, 1542, 1632, 1920, 2040, 2048, 2056, 2176, 2560, 2570, 2720, 3072, 3084, 3264, 3840, 3855, 4080, 4096, 4112, 4352, 4369, 5120, 5140, 5440, 6144, 6168, 6528, 7680, 7710, 8160, 8192, 8224, 8704, 8738, 10240, 10280, 10880, 12288, 12336, 13056, 13107, 15360, 15420, 16320, 16384, 16448, 17408, 17476, 20480, 20560, 21760, 21845, 24576, 24672, 26112, 26214, 30720, 30840, 32640, 32768, 32896, 34816, 34952

Let's take 20 as an example. The numbers that are coprime to 20 are 1, 3, 7, 9, 11, 13, 17, 19 and so we have we have a count of 8:$$ \phi(20)=8=2^3$$Next, let's consider the sum of the anti-divisors of a number. First let's recall what an antidivisor is using Gemini to explain in very clear terms:

An anti-divisor is a number that fails to divide a target integer by the largest possible margin. While a regular divisor divides a number evenly with no remainder, an anti-divisor leaves the most unbiased, centered remainder possible. [1, 2, 3]

The concept was first formalized by mathematician Jon Perry. [1]

Mathematical Rules

An integer \(d\) (where \(1 < d < n\)) is an anti-divisor of \(n\) if it satisfies one of the following rules: [1, 2]

• For even values of \(d\): The number \(d\) divides \(2n\) evenly. Mathematically, we have:$$n \bmod d = \dfrac{d}{2}$$
• For odd values of \(d\): The number \(d\) divides \(2n - 1\) or \(2n + 1\) evenly. Mathematically we have  [1, 2]:$$n \bmod d = \dfrac{d-1}{2}  \text{ or } n \bmod d = \dfrac{d+1}{2}$$

Visual Example: Anti-divisors of 20

To find the anti-divisors of \(20\), you look for numbers between \(2\) and \(19\) that fail to divide \(20\), but cleanly divide \(40\) (twice \(20\)) or \(39\)/\(41\). [1]

• 3 is an odd anti-divisor because \(3 \times 13 = 39\), leaving a gap of \(1\) (since \(\frac{3-1}{2} = 1\), and \(20 \bmod 3 = 2\)).
• 8 is an even anti-divisor because \(8 \times 5 = 40\). It divides \(2 \times 20\) perfectly.
• 13 is an odd anti-divisor because \(13 \times 3 = 39\). [1]

Key Facts

• The number 1 is never an anti-divisor, as it evenly divides all integers.
• The number 2 has no anti-divisors.
• Prime numbers have a limited number of anti-divisors based heavily on multiples of \(2n\). [1, 2, 3]

For a complete look at integer sequences and properties, you can explore the The Anti-Divisor - OEIS catalog. [1]

There are 20 numbers in the range up to 40000 that satisfy and these are (permalink):

6, 8, 9, 14, 36, 89, 96, 221, 541, 576, 740, 778, 1854, 2114, 2571, 10277, 13631, 16160, 16389, 39428

Let's take 39428 as an example. The antidivisors of this number are:

3, 5, 7, 8, 15, 21, 35, 105, 751, 2253, 3755, 5257, 11265, 15771, 26285

The sum of these anti-divisors is \(65536 = 2^{16}\) and so it is perfect power.

Further investigation could target:

• the sum of the proper divisors
• the sum of the non-divisors
• the determinant of the circulant matrix
• the arithmetic derivative

Beatty's Theorem

This video on Beatty's Theorem by Euclidia is very interesting and very surprising. It shows that the integers can be split into two disjoint sets using an irrational number of your choice provided it is greater than 1. Each pair is unique to the irrational number being used. The two infinite sets cover the entire range of positive integers.

It's extremely easy to generate the two sequences using the following SageMath code. Here, using \( \sqrt{2} \), I've generated the first 25 terms of each sequence:

a=sqrt(2)
b=a/(a-1)
A,B=[],[]
for n in [1..25]:
    A.append(floor(n*a))
    B.append(floor(n*b))
print(A)

print(B)

[1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 24, 25, 26, 28, 29, 31, 32, 33, 35]
[3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, 47, 51, 54, 58, 61, 64, 68, 71, 75, 78, 81, 85] 

Using different values of \(a\) such as \( \pi \) or \(e\) yields different disjoint sets:

\( \pi\) yields the following sequences:

[3, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, 37, 40, 43, 47, 50, 53, 56, 59, 62, 65, 69, 72, 75, 78]

[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36]

\(e\) yields the following sequences:

[2, 5, 8, 10, 13, 16, 19, 21, 24, 27, 29, 32, 35, 38, 40, 43, 46, 48, 51, 54, 57, 59, 62, 65, 67]

[1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, 30, 31, 33, 34, 36, 37, 39]

INTERACTIVE LINK

Here a link to an interactive report that Gemini created regarding Beatty's sequences. I struggled to get this to work in Blogger but by using this external link, it all worked fine. Something to remember in the future.

Palindromic Day 28182

Palindromic properties of 28182 (showing only sequence members up to 40000):

A098834: palindromic Smith numbers.

4, 22, 121, 202, 454, 535, 636, 666, 1111, 1881, 3663, 7227, 7447, 9229, 10201, 17271, 22522, 24142, 28182, 33633, 38283

A Smith number is a composite number where the sum of its digits equals the sum of the digits of its prime factors. For 28182:$$ \begin{align} 28182 &\rightarrow 2+8+1+8+2 = 21 \\ 28182 &= 2 \times 3 \times 7 \times 11 \times 61\\ &\rightarrow 2 + 3 + 7 + 1+1+6 +1 =21 \end{align}$$

A046395: palindromes that are the product of 5 distinct primes.

6006, 8778, 20202, 28182

Here \(28182 = 2 \times 3 \times 7 \times 11 \times 61 \)

A099052: all palindromes of length > 1 in the decimal expansion of \(e\).

\(e\) = 2.71828182845904523536028747135266 ...

A045571: numbers that are palindromic, divisible by 11 and have an odd number of digits.

121, 242, 363, 484, 616, 737, 858, 979, 10901, 11011, 12221, 13431, 14641, 15851, 17171, 18381, 19591, 20702, 21912, 22022, 23232, 24442, 25652, 26862, 28182, 29392, 30503, 31713, 32923, 33033, 34243, 35453, 36663, 37873, 39193

All the palindromic numbers with an even number of digits are divisible by 11. The number of palindromic numbers with \(2k+1\) digits that are divisible by 11 is \((10^{k+1} + (-1)^k)/11\), and their asymptotic relative density within the set of all palindromic numbers with an odd number of digits is 1/11 (from OEIS comments).

A113838: palindromes sandwiched between twin primes.

4, 6, 282, 828, 858, 2112, 21012, 21612, 23832, 26262, 26862, 28182

Here of course the twin primes are 28181 and 28183.

A032751: palindromic Super-3 Numbers.

4554, 6776, 17471, 22322, 22722, 28182

Super-3 numbers \(n\) are of the form \(3 \times n^3 \) and contain three consecutive 3's.

Here \(3 \times 28182^3 = 67148557\textbf{333}704\)