28224: An Interesting Number

28224 has 194 entries in the OEIS which is extraordinarily high for a five digit number. In this post I'll be discussing some of this number's most interesting properties but not all of them. There are just too many. It's prime factorisation is:$$28224=2^6 \times 3^2 \times 7^2$$FIRST INTERESTING PROPERTY

Numbers that are perfect squares are quite rare in the range up to 40000. There are only 200 of them and 28224, my diurnal age today, is one of them. It has the property that:$$28224=168^2$$The number of days between my experience of them is a little less than a year. There is a gap of exactly 365 days between \(183^2\) and \(182^2\) since:$$ \begin{align} 183^2-182^2 &= (183 + 182)(183-182) \\ &=365 \times 1 \\ &=365 \end{align}$$I'll be \(33124\) or \(182^2\) days when I'm over \(90\) years old so I may not get to experience the transition from this square to the next.

28224 is also a Loeschian number since it is equal to \(72^2+ 72 \times 120 + 120^2\).

28224 also has a product of digits (256) that is a perfect square since \(256=16^2\).

SECOND INTERESTING PROPERTY

Numbers that are the sum of two positive cubes are relatively rare in the range up to 40000. In fact, there are only 378 numbers in the range up to 40000 and 28224 is one of them because:$$28224=22^3 + 26^3$$These numbers form OEIS A004999.

THIRD INTERESTING NUMBER

Energetic numbers are numbers that can be broken into two or more substrings and expressed as a sum of (possibly different) positive powers of those substrings. They form OEIS  A055480. 28224 is one such number because:$$28224=28^3 + 2^{11} + 2^7 + 4^6$$I discuss this category of numbers in my blog post Energetic Numbers.

FOURTH INTERESTING NUMBER

Friedman numbers are positive integers which can be written in some non-trivial way using its own digits, together with the symbols + – × / ^ ( ) and concatenation. 28224 is one such number because:$$28224 = (2 + 82)^2 × 4$$It is said to be a "nice" Friedman number because the digits are in the same order as the number. These numbers are listed on my blog post Narcissistic, D-Powerfull and Friedman Numbers.

FIFTH INTERESTING NUMBER

28224 has the property that certain of its factors (not necessarily prime) can be arranged to form a palindrome. Specifically:$$2 \times 2 \times 2 \times 882 \times 2 \times 2 = 22288222$$I've written about these sorts of numbers in a post titled Why Is 313131 An Interesting Number?

SIXTH INTERESTING PROPERTY

28224 is a concatenation of powers of 2 since:$$28224= 2^1 \; || \; 2^3 \; || \; 2^1 \; || \; 2^1 \; || \; 2^2$$I've written about numbers that can be formed in this way in a blog post titled Nothing New Under The Sun. It is also a concatenation of multiples of 7 since:$$28224= (7 \times 4) \, || \, (7 \times 32)$$I posted about these sorts of concatenations in my blog post More Numbers as Concatenations.

SEVENTH INTERESTING PROPERTY

28224 is a member of OEIS A253824 where$$ \text{numbers } m = s \, || \, t \text{ such that } m = \sigma(s) \times \sigma(t)$$where || represents concatenation. In the case of 28224 we have:$$ \begin{align} 28224 &= 28 \, || \, 224 \\ &= \sigma(28) \times \sigma(224) \\ &= 56 \times 504 \\ &=28224 \end{align}$$28224 is only the third such number in the range up to 40000. The two earlier numbers are 540 and 2352.

EIGHTH INTERESTING PROPERTY

28224 has a digit sum of 18 and when this is added to the number the result is 28242 which has the same digits as 28224 but in a slightly different order. This property makes it a member of OEIS A246420.

A246420

Numbers \(n\) such that \(n\)  + digit sum of \(n\) is a permutation of the decimal digits of \(n\) .

More Numbers as Concatenations

In my previous post, Divisors and Antidivisors: A Fresh Perspective, I dealt with numbers formed by concatenations of divisors and also by antidivisors of a number. For example, consider the number 15:

• 15 has divisors of 1, 3, 5 and 15
• 13515 is formed by concatenating these divisors
• 15 divides 13515 to give 901
• numbers with this property belong to OEIS A069872
• 15 has proper divisors of 1, 3 and 5
• 135 is formed by concatenating these proper divisors
• 15 divides 135 to give 9
• numbers with this property belong to OEIS A240265
• 15 has antidivisors of 2, 6 and 10
• 2610 is formed by concatenating these antidivisors
• 15 divides 2610 to give 174
• numbers with this property belong to OEIS A249764

In an earlier post titled, Nothing New Under The Sun, I looked at numbers formed by concatenation of powers of prime digits. Let's take 128864 which can be formed by a concatenation of powers of 2:$$128864=2^7 \, | \, 2^3 \, | \, 2^6$$where | is the symbol commonly used for concatenation. Numbers like this belong to OEIS A381259. 

In this post I want to look at numbers that are a concatenation of the multiples of a digit but that do not contain the digit itself. Let's take 28133 as an example. It's not immediately obvious but this number can be broke into two parts, 28 and 133, both of which are multiples of 7:$$28133 \rightarrow 28 \, | \, 133 = (7 \times 4) \, | \, (7 \times 19)$$There are 190 such numbers in the range up 40000. They are (permalink):

1414, 1421, 1428, 1435, 1442, 1449, 1456, 1463, 1484, 1491, 1498, 2121, 2128, 2135, 2142, 2149, 2156, 2163, 2184, 2191, 2198, 2828, 2835, 2842, 2849, 2856, 2863, 2884, 2891, 2898, 3535, 3542, 3549, 3556, 3563, 3584, 3591, 3598, 4242, 4249, 4256, 4263, 4284, 4291, 4298, 4949, 4956, 4963, 4984, 4991, 4998, 5656, 5663, 5684, 5691, 5698, 6363, 6384, 6391, 6398, 8484, 8491, 8498, 9191, 9198, 9898, 14105, 14112, 14119, 14126, 14133, 14140, 14154, 14161, 14168, 14182, 14189, 14196, 14203, 14210, 14224, 14231, 14238, 14245, 14252, 14259, 14266, 14280, 14294, 14301, 14308, 14315, 14322, 14329, 14336, 14343, 14350, 21105, 21112, 21119, 21126, 21133, 21140, 21154, 21161, 21168, 21182, 21189, 21196, 21203, 21210, 21224, 21231, 21238, 21245, 21252, 21259, 21266, 21280, 21294, 21301, 21308, 21315, 21322, 21329, 21336, 21343, 21350, 28105, 28112, 28119, 28126, 28133, 28140, 28154, 28161, 28168, 28182, 28189, 28196, 28203, 28210, 28224, 28231, 28238, 28245, 28252, 28259, 28266, 28280, 28294, 28301, 28308, 28315, 28322, 28329, 28336, 28343, 28350, 35105, 35112, 35119, 35126, 35133, 35140, 35154, 35161, 35168, 35182, 35189, 35196, 35203, 35210, 35224, 35231, 35238, 35245, 35252, 35259, 35266, 35280, 35294, 35301, 35308, 35315, 35322, 35329, 35336, 35343, 35350

Some of these numbers find themselves in the OEIS for reasons that involve multiples but in slightly different ways. Take 28168, listed above, as an example. It is a member of OEIS A009440: a(\(n\)) is the concatenation of \(n\) and 6\(n\). Thus we have:$$ \begin{align} 28168 &= 28 | (6 \times 28) \\28168 &= (7 \times 4) | (7 \times 24) \end{align}$$Choosing multiples of 2 and 3 produce 641 and 455 suitable numbers respectively while choosing multiples of 5 produces 78 suitable numbers in the range up to 40000 (permalink):

1010, 1020, 1030, 1040, 1060, 1070, 1080, 1090, 2020, 2030, 2040, 2060, 2070, 2080, 2090, 3030, 3040, 3060, 3070, 3080, 3090, 4040, 4060, 4070, 4080, 4090, 6060, 6070, 6080, 6090, 7070, 7080, 7090, 8080, 8090, 9090, 10100, 10110, 10120, 10130, 10140, 10160, 10170, 10180, 10190, 10200, 10210, 10220, 10230, 10240, 20100, 20110, 20120, 20130, 20140, 20160, 20170, 20180, 20190, 20200, 20210, 20220, 20230, 20240, 30100, 30110, 30120, 30130, 30140, 30160, 30170, 30180, 30190, 30200, 30210, 30220, 30230, 30240

In the case of 1010, we have:$$1010 \rightarrow 10 \, | \, 10 = (5 \times 2) \, | \, (5 \times 2)$$The permalink allows experimentation with other digits or even numbers. There's no deep Mathematics in all this just another way to spot patterns in numbers.

Average of Three Primes Squared

The number associated with my diurnal age today is 28129 and it is a member of OEIS A075893:

     A075893: average of three successive primes squared with \(n > 3\) 
                

    \( \dfrac{\text{prime} (n)^2+ \text{prime}(n+1)^2+\text{prime}(n+2)^2)}{3}\)

The initial terms are:

65, 113, 193, 273, 393, 577, 777, 1057, 1337, 1633, 1913, 2289, 2833, 3337, 3897, 4417, 4953, 5537, 6153, 7017, 8073, 9177, 10073, 10753, 11313, 12033, 13593, 15353, 17353, 18417, 20097, 21441, 23217, 24673, 26369, 28129, 29953, 31577, 33761, 35497, 37513, 38553

The OEIS comments state that:

Unlike the average of three successive primes, the average of three successive primes (greater than 3) squared is always integral.

I asked Gemini why this was so:

The reason the average of the squares of three successive primes (greater than 3) is always an integer is based on a fascinating pattern about how primes behave when divided by 3.

In fact, this rule doesn't just apply to successive primes—it applies to any three primes greater than 3. Here is the step-by-step mathematical breakdown of why this happens.

1. All Primes Greater Than 3 Avoid Multiples of 3

By definition, a prime number is only divisible by 1 and itself. This means that any prime number strictly greater than 3 cannot be a multiple of 3.

If we divide any number by 3, we can only ever get one of three possible remainders: 0, 1, or 2.

• Because primes greater than 3 are not divisible by 3, they will never have a remainder of 0.
• Therefore, any prime number p > 3 must leave a remainder of either 1 or 2 when divided by 3.

In algebra, this means every prime p > 3 can be written in one of two ways, where k is an integer:

• p = 3k + 1 (leaves a remainder of 1)
• p = 3k + 2 (leaves a remainder of 2, which can also be written as p = 3k - 1)

2. Squaring the Primes

Now, let's look at what happens when we square these prime numbers.

Case A: The prime is 3k + 1
p² = (3k + 1)²
p² = 9k² + 6k + 1

We can factor out a 3 from the first two terms:
p² = 3(3k² + 2k) + 1

Because 3(3k² + 2k) is a multiple of 3, the entire expression leaves a remainder of 1 when divided by 3.

Case B: The prime is 3k - 1
p² = (3k - 1)²
p² = 9k² - 6k + 1

Again, factor out a 3 from the first two terms:
p² = 3(3k² - 2k) + 1

Just like the first case, this leaves a remainder of 1 when divided by 3.

The Key Rule: No matter which type of prime you have, as long as it is greater than 3, its square will always be exactly 1 more than a multiple of 3.

3. Averaging Three Squared Primes

Let p₁, p₂, and p₃ be any three primes greater than 3. Based on our rule above, we can express their squares as:

• p₁² = 3a + 1
• p₂² = 3b + 1
• p₃² = 3c + 1

Now, add them together to find the sum:
Sum = (3a + 1) + (3b + 1) + (3c + 1)
Sum = 3a + 3b + 3c + 3

We can factor out a 3 from the entire sum:
Sum = 3(a + b + c + 1)

Since the sum of the three squared primes is a perfect multiple of 3, dividing that sum by 3 to find the average will always result in a clean integer:
Average = 3(a + b + c + 1) / 3 = a + b + c + 1

Because a, b, and c are integers, the average itself is guaranteed to be an integer.

A Truly Incredible Fact About The Number 37

It was this video from the YouTube channel Veritasium that made me aware of the considerable interest attached to the number 37.

I then found this post from a blogger, Chris Grossack, to be especially helpful in explaining the following:

37 is the median value for the second prime factor of an integer; thus the probability that the second prime factor of an integer chosen at random is smaller than 37 is approximately 50%.

He also uses SageMath for his calculations which was an added bonus. Here is the permalink to the calculation to determine the median using the first 100,000 numbers. The output is shown in Figure 1.

Figure 1

The actual proof is summarised in the information contained in Figure 2 which is more than I can comprehend, but I'll include it here:

Figure 2

The blogger uses the formulae shown in Figure 2 to once again show that 37 is the median value. Here is the permalink and the output is shown in Figure 3.

Figure 3

Of course this is not 37's only claim to fame. Wikipedia has an entry for the number 37 and some of the interesting facts contained in that article include a 3 x 3 magic square with 37 at its centre. See Figure 4.

Figure 4

Its magic constant is 37 x 3 = 111, where 3 and 37 are the first and third base-ten unique primes (the second such prime is 11). I wasn't familiar with the notion of a unique prime and so I'll include a definition from the Wikipedia article here:

A prime \(p\) (where \(p\) ≠ 2, 5 when working in base 10) is called unique if there is no other prime \(q\) such that the period length of the decimal expansion of its reciprocal, 1/\(p\), is equal to the period length of the reciprocal of \(q\), 1/\(q\). For example, 3 is the only prime with period 1, 11 is the only prime with period 2, 37 is the only prime with period 3, 101 is the only prime with period 4, so they are unique primes. The next larger unique prime is 9091 with period 10, though the next larger period is 9 (its prime being 333667). Unique primes were described by Samuel Yates in 1980.

I've written about 37 extensively as well in a post titled Star Numbers from the 7th of June 2019. There is a website dedicated to the number 37. It's mentioned by its creator in the YouTube video earlier but, as he himself admits, it hasn't been updated in very many years. However, it still contains a wealth of information.

For example, the site describes a method of determining if a number is divisible by 37. This is the method:

• Divide the number up in groups of three digits, starting from the right.
  (The left-most group may not have three digits.)
• Add the groups together.
• Repeat steps 1 and 2 if the result is still longer than three digits, repeat steps 1 and 2.
• Examine the final three-digit (or smaller) number

The original number is divisible by 37 if and only if this three-digit number is.

I often take note of car number plates here in Jakarta. These are typically of the form B-xxxx where xxxx is a four digit number. It's easy to determine if the four digit number is divisible by 3 because the first digit is simply added to the remaining three. Using leading zeros, the multiples of 37 are:

037, 074, 111, 148, 185, 222, 259, 296, 333, 370, 407, 444, 481, 518, 555, 592, 629, 666, 703, 740, 777, 814, 851, 888, 925, 962, 999

The repeated digit numbers (111 to 999) are a dead given away but the others are not two difficult to identify. Let's consider a number plate like B-1258. The 1258 --> 1 + 258 = 259 = 7 x 37. In this case, the 7 can be divided in to reveal the 37 rather than dealing with division by 37. There is no limit to what can be said about the number 37 but at least in this post and my earlier post of star numbers I've made a start.

Arithmetic Derivative Records

On June 19th 2019 I made a post titled Arithmetic Derivative and in this post I'll return to the topic. I was prompted to do so by one of the properties of the number associated with my diurnal age today. The number is 27265 and it has the property that its arithmetic derivative, 11448, has no digits in common with the number itself. I've listed these numbers in an entry in my Bespoken for Sequences.

I'm not going to pursue that topic in this post but I was reminded of arithmetic derivatives and got to thinking about records being set by the size of arithmetic derivatives as the natural numbers are traversed. It didn't take long to develop a SageMath algorithm to explore this topic (permalink). Detailed results are shown below in Table 1 with derivatives up 40000 in size, although the algorithm lists the records for numbers up to one million.

Table 1

The record setting sizes of the arithmetic derivatives form OEIS A131116 and the initial values are as follows:

4, 5, 12, 16, 32, 44, 80, 112, 192, 272, 448, 640, 1024, 1472, 2304, 2368, 3328, 3392, 5120, 5376, 7424, 7744, 11264, 12032, 16384, 16640, 17408, 24576, 26624, 35840, 36864, 38656, 53248, 58368, 77824, 80896, 84992, 114688, 126976, 167936, 176128, 185344, 245760, 274432, 360448, 380928, 401408, 524288, 528384, 589824, 593920, 770048, 774144, 819200, 864256, 1114112, 1130496, 1261568, 1277952, 1638400, 1658880, 1753088, 1851392, 2359296, 2408448, 2686976, 2736128, 3473408, 3538944, 3735552, 3948544, 4980736, 5111808, 5701632, 5832704, 7340032, 7520256, 7929856, 8388608

Table 2 shows a graph of these numbers:

Table 2

As can be seen from Table 1, all the numbers associated with these records contain all powers of 2 together with multiples of these powers. The initial numbers are:

4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 640, 768, 960, 1024, 1280, 1536, 1920, 2048, 2560, 3072, 3584, 3840, 4096, 5120, 6144, 7168, 7680, 8192, 10240, 12288, 14336, 15360, 16384, 20480, 24576, 28672, 30720, 32768, 40960, 49152, 57344, 61440, 65536, 73728, 81920, 90112, 98304, 110592, 114688, 122880, 131072, 147456, 163840, 180224, 196608, 221184, 229376, 245760, 262144, 294912, 327680, 360448, 393216, 442368, 458752, 491520, 524288, 589824, 655360, 720896, 786432, 884736, 917504, 983040

Divisor Runs

 My diurnal age today, 26885, is a member of OEIS A006601:

A006601

Numbers \(k\) such that \(k\), \(k+1\), \(k+2\) and \(k+3\) have the same number of divisors.

This means that 26885, 26886, 26887, 26888 and 26889 have the same number of divisors. Let's check that this is the case:

26885 = 5 x 19 x 283 and has 8 divisors

26886 = 2 x 3 x 4481 and has 8 divisors

26887 = 7 x 23 x 167 and has 8 divisors

26888 = 2^3 x 3361 and has 8 divisors

Let's not forget the rule for determining the number of divisors from the factorisation: add one to the index of each prime factor and then multiply them together. Runs of four numbers with the same number of divisors are rare. 

Below are listed the numbers up to one million, all members of the OEIS sequence (permalink). There are only 1171 of them, representing 0.1171% of the numbers in the range. 

242, 3655, 4503, 5943, 6853, 7256, 8392, 9367, 10983, 11605, 11606, 12565, 12855, 12856, 12872, 13255, 13782, 13783, 14312, 16133, 17095, 18469, 19045, 19142, 19143, 19940, 20165, 20965, 21368, 21494, 21495, 21512, 22855, 23989, 26885, 28135, 28374, 28375, 28376, 29605, 30583, 31735, 31910, 32005, 32792, 33062, 33608, 33845, 34069, 36392, 37256, 40311, 40312, 41335, 42805, 42806, 43304, 43526, 43766, 44213, 45686, 45733, 47845, 48054, 49147, 49765, 50582, 50583, 51752, 54103, 54585, 54966, 55063, 55254, 55255, 55976, 56343, 58952, 59815, 60231, 60232, 60663, 60664, 61142, 62343, 65334, 66952, 67015, 68104, 69303, 71095, 73927, 74053, 76262, 76982, 77432, 78535, 78872, 79094, 79095, 79591, 80726, 82855, 84469, 86887, 87655, 87656, 87896, 90181, 90182, 90183, 91495, 93063, 94262, 94645, 95384, 95414, 95512, 95845, 95846, 97255, 98102, 98984, 99655, 99656, 100711, 100952, 101125, 103352, 103621, 103622, 104222, 104744, 104870, 105301, 107365, 108902, 109191, 109765, 109766, 112567, 113942, 115591, 115592, 115912, 116965, 117032, 118069, 118261, 118615, 118923, 120727, 120728, 120965, 121045, 121046, 122151, 122152, 122871, 122872, 123944, 124663, 125335, 129829, 130135, 131815, 133624, 134582, 136375, 136825, 139863, 141654, 142454, 142455, 142806, 142807, 143365, 145352, 146936, 151285, 152102, 152552, 152630, 152631, 153461, 153543, 153703, 153992, 157493, 157494, 157495, 157910, 158216, 158342, 160934, 162295, 164982, 165542, 166791, 167671, 169112, 169141, 169813, 171893, 171894, 171895, 171896, 172501, 173893, 173912, 174054, 174055, 174872, 175143, 175144, 178086, 178087, 180901, 180902, 180965, 180966, 180967, 180968, 181207, 182215, 183205, 183206, 183554, 183685, 184327, 184328, 185815, 186231, 186951, 186952, 188293, 188294, 191943, 191944, 192728, 194695, 197463, 198776, 198806, 199111, 199112, 199255, 199733, 199832, 201512, 201943, 203432, 203621, 204323, 204324, 204806, 206390, 206552, 210133, 210134, 210135, 211592, 212005, 212006, 214885, 217144, 217526, 218695, 219063, 220232, 220405, 223687, 223861, 223976, 224005, 224776, 225463, 225464, 226792, 227576, 229015, 229352, 229685, 230167, 230869, 231413, 233942, 234615, 235112, 235495, 235496, 236006, 236245, 236246, 236792, 237031, 237063, 238694, 240904, 243415, 243445, 244742, 246294, 246631, 246632, 248311, 248504, 249445, 249512, 251751, 251752, 252967, 253014, 253015, 258008, 259205, 261734, 261992, 262661, 262932, 263765, 263766, 264296, 264952, 266485, 266869, 267655, 268021, 268933, 269912, 270181, 270613, 271526, 272455, 273368, 274088, 277543, 277624, 279302, 279542, 280230, 280712, 282055, 282245, 282584, 283496, 284581, 285413, 287029, 287704, 287814, 288085, 290981, 291781, 292853, 293029, 294085, 295014, 295015, 295352, 296453, 298694, 298695, 298696, 298903, 299863, 301062, 301189, 301622, 302103, 304262, 304675, 305335, 306086, 306631, 306632, 306805, 307254, 307255, 307285, 307445, 308342, 309415, 309416, 309655, 309656, 309830, 310503, 310808, 312151, 312152, 316326, 319765, 321304, 321543, 322214, 324662, 326791, 326792, 328262, 328263, 328933, 329125, 329144, 329432, 331285, 331638, 332102, 332103, 332888, 333176, 334085, 335093, 335863, 336485, 337254, 337335, 337431, 341030, 341031, 341463, 341605, 343189, 343623, 344935, 345782, 346502, 346503, 346504, 350312, 352135, 352983, 353605, 353606, 354056, 355453, 356293, 356504, 358015, 358309, 358934, 360055, 360183, 361275, 361285, 361322, 362341, 362534, 363207, 365269, 365384, 366005, 366535, 369061, 369062, 369063, 369127, 370165, 372711, 373143, 373784, 374630, 374888, 376453, 376742, 376743, 376744, 377095, 377896, 379255, 379670, 379862, 380344, 381592, 381702, 382854, 383872, 384391, 386726, 386965, 386966, 387302, 387703, 390470, 390533, 393589, 393653, 394454, 395462, 395704, 396902, 397352, 398408, 399368, 399655, 399656, 400165, 400166, 400375, 400693, 400694, 401270, 401653, 402294, 402295, 403526, 404391, 404392, 406135, 406136, 407605, 411062, 412615, 413176, 414421, 417445, 417446, 417608, 418135, 420005, 420006, 422885, 422886, 423031, 424712, 425576, 426584, 427015, 427352, 427494, 428053, 428392, 430645, 431509, 431895, 432085, 432086, 432392, 433064, 434552, 434584, 435016, 437671, 439016, 440869, 445062, 446295, 446296, 447365, 447366, 448903, 451333, 451733, 451784, 451910, 452215, 453895, 455365, 456952, 459734, 460742, 460805, 461511, 461512, 461671, 464214, 464582, 464583, 464871, 464872, 465031, 465032, 465544, 465589, 467030, 467031, 468902, 469864, 471367, 472070, 472853, 475736, 475976, 476377, 479845, 479846, 480134, 480664, 481061, 482693, 482744, 483941, 486229, 486487, 486962, 487191, 487192, 490375, 490855, 493382, 493383, 494744, 494934, 495590, 496792, 500312, 501845, 501846, 502183, 506245, 506821, 506905, 507125, 507782, 507783, 507944, 508712, 515192, 517304, 517431, 517432, 519366, 519590, 520087, 520088, 520807, 521126, 521461, 525064, 525845, 525846, 528853, 528902, 530168, 530870, 530887, 531445, 531446, 532741, 532742, 532983, 533703, 533767, 535064, 536167, 536583, 538374, 538453, 539270, 539911, 539912, 541045, 542792, 543062, 543063, 545912, 547192, 548536, 548776, 549590, 549992, 551192, 552390, 554741, 554742, 555032, 555302, 556743, 556869, 558086, 558229, 558230, 559013, 559445, 559688, 560791, 560792, 562328, 564008, 564053, 565014, 567302, 569864, 570053, 570533, 571621, 571863, 572214, 572215, 572405, 572744, 573365, 574743, 574885, 574886, 576895, 579062, 580935, 580982, 581509, 583286, 583383, 585494, 586885, 586952, 587462, 587864, 588485, 588728, 589208, 589592, 589765, 590935, 592376, 594344, 596071, 596245, 596533, 597736, 598087, 600294, 604262, 605432, 606343, 607526, 610310, 610311, 610312, 610904, 611384, 613192, 614965, 615061, 615352, 616063, 616135, 616375, 617096, 617125, 617896, 619722, 619832, 621445, 624054, 627654, 627735, 628664, 628855, 628856, 631432, 631832, 632694, 632695, 633110, 634232, 634262, 636710, 638407, 638408, 639734, 639991, 639992, 640855, 640885, 640886, 641765, 642295, 644293, 645031, 645205, 646232, 646789, 647381, 647382, 647383, 647384, 648326, 648344, 649623, 649624, 649830, 649856, 650245, 650741, 652885, 654182, 654952, 656312, 656934, 657542, 657895, 658232, 658711, 659701, 662390, 662485, 664183, 664184, 665093, 665816, 668211, 668632, 669736, 672536, 676165, 676262, 676711, 678486, 679735, 679736, 680726, 680869, 682070, 684805, 685864, 688134, 689143, 689432, 690805, 693031, 693032, 693542, 693685, 694504, 694741, 694742, 697206, 700134, 700135, 701463, 701544, 701911, 701912, 702182, 702183, 703335, 705416, 706069, 707032, 707286, 707287, 707288, 709303, 710454, 711752, 712405, 712453, 713845, 714632, 716005, 716006, 718232, 718471, 720085, 720245, 720246, 722055, 723991, 723992, 725845, 726565, 726566, 727094, 729542, 729543, 729544, 730645, 731461, 731462, 731703, 731941, 733158, 734696, 734948, 736070, 738902, 740310, 741254, 741783, 742552, 742567, 742855, 744005, 744709, 744806, 751142, 751303, 754743, 755191, 755192, 755462, 756343, 756581, 757207, 757544, 757670, 760504, 760711, 760712, 760855, 762086, 762296, 763688, 763734, 763735, 763862, 765032, 765894, 766263, 766335, 766982, 767431, 767893, 768776, 769189, 769862, 769863, 769864, 770693, 771445, 771446, 772213, 772214, 772711, 776936, 777254, 777271, 777415, 778294, 778981, 780854, 781832, 782312, 782742, 783416, 783894, 783895, 786565, 786566, 787735, 787736, 788504, 788583, 789832, 789895, 792326, 793192, 796405, 796502, 796503, 798470, 799255, 802086, 802855, 802856, 802904, 803605, 805862, 807079, 807445, 809301, 809302, 809462, 810902, 810903, 812215, 812821, 813942, 813943, 817592, 820262, 820263, 820310, 820471, 820856, 822151, 823285, 823432, 823862, 824629, 824744, 825735, 828872, 829445, 829622, 832502, 832503, 833192, 836389, 836390, 836407, 836632, 837543, 838405, 838615, 839030, 839991, 839992, 841143, 841592, 842005, 842006, 842101, 842373, 844133, 844231, 844232, 845221, 845382, 845383, 845815, 846744, 846902, 846966, 846967, 847254, 847255, 848246, 848582, 848583, 848695, 848966, 849493, 849494, 850088, 852296, 853095, 853096, 854018, 855320, 855703, 857606, 857814, 857815, 858470, 862616, 863528, 864184, 865013, 865526, 866341, 866342, 868567, 868614, 869096, 871381, 871382, 875095, 875462, 876152, 876295, 876296, 876631, 877928, 880119, 880662, 882183, 883351, 883352, 883832, 884504, 884821, 885512, 887576, 889189, 889816, 890534, 891271, 891415, 891776, 893126, 893512, 894952, 896485, 898328, 898374, 898645, 898885, 901503, 903351, 903512, 904405, 904712, 905511, 905512, 905815, 906967, 907464, 908631, 908632, 909176, 911270, 911912, 915781, 915896, 916373, 917527, 919207, 923989, 927624, 928232, 930181, 930390, 934311, 934312, 937025, 938203, 939416, 939894, 940312, 941432, 942485, 943285, 943815, 944407, 944485, 945783, 946712, 947703, 947767, 948965, 950005, 950342, 950343, 951205, 952039, 952552, 954392, 956552, 957414, 957512, 958645, 958791, 958792, 959365, 959846, 960967, 960968, 961301, 961494, 962965, 963445, 964310, 969832, 970616, 970808, 980184, 981205, 984135, 984295, 984344, 986455, 987032, 988374, 988375, 988952, 989095, 989894, 990661, 990662, 991045, 991622, 991910, 992693, 992870, 993542, 995815, 995942, 996229, 998389, 999649

Naturally I was interested in finding the limit of these runs from one to one million. What about runs of five numbers? Well the number shrinks drastically to just 179 (permalink).

11605, 12855, 13782, 19142, 21494, 28374, 28375, 40311, 42805, 50582, 55254, 60231, 60663, 79094, 87655, 90181, 90182, 95845, 99655, 103621, 109765, 115591, 120727, 121045, 122151, 122871, 142454, 142806, 152630, 157493, 157494, 171893, 171894, 171895, 174054, 175143, 178086, 180901, 180965, 180966, 180967, 183205, 184327, 186951, 188293, 191943, 199111, 204323, 210133, 210134, 212005, 225463, 235495, 236245, 246631, 251751, 253014, 263765, 295014, 298694, 298695, 306631, 307254, 309415, 309655, 312151, 326791, 328262, 332102, 341030, 346502, 346503, 353605, 369061, 369062, 376742, 376743, 386965, 399655, 400165, 400693, 402294, 404391, 406135, 417445, 420005, 422885, 432085, 446295, 447365, 461511, 464582, 464871, 465031, 467030, 479845, 487191, 493382, 501845, 507782, 517431, 520087, 525845, 531445, 532741, 539911, 543062, 554741, 558229, 560791, 572214, 574885, 610310, 610311, 628855, 632694, 638407, 639991, 640885, 647381, 647382, 647383, 649623, 664183, 679735, 693031, 694741, 700134, 701911, 702182, 707286, 707287, 716005, 720245, 723991, 726565, 729542, 729543, 731461, 755191, 760711, 763734, 769862, 769863, 771445, 772213, 783894, 786565, 787735, 796502, 802855, 809301, 810902, 813942, 820262, 832502, 836389, 839991, 842005, 844231, 845382, 846966, 847254, 848582, 849493, 853095, 857814, 866341, 871381, 876295, 883351, 905511, 908631, 934311, 950342, 958791, 960967, 988374, 990661

What about runs of six numbers? There are just 18 (permalink).

28374, 90181, 157493, 171893, 171894, 180965, 180966, 210133, 298694, 346502, 369061, 376742, 610310, 647381, 647382, 707286, 729542, 769862

What about runs of seven numbers? There are a mere three (permalink) and there are no runs of eight numbers in the range up to one million.

171893, 180965, 647381

Let's look at the factorisation of these numbers. We find all three have eight divisors and most are sphenic numbers except for a single number that is divisible by eight.

171893 = 19 x 83 x 109 and has 8 divisors

171894 = 2 x 3 x 28649 and has 8 divisors

171895 = 5 x 31 x 1109 and has 8 divisors

171896 = 2^3 x 21487 and has 8 divisors

171897 = 3 x 11 x 5209 and has 8 divisors

171898 = 2 x 61 x 1409 and has 8 divisors

171899 = 7 x 13 x 1889 and has 8 divisors

180965 = 5 x 17 x 2129 and has 8 divisors

180966 = 2 x 3 x 30161 and has 8 divisors

180967 = 37 x 67 x 73 and has 8 divisors

180968 = 2^3 x 22621 and has 8 divisors

180969 = 3 x 179 x 337 and has 8 divisors

180970 = 2 x 5 x 18097 and has 8 divisors

180971 = 7 x 103 x 251 and has 8 divisors

647381 = 7 x 23 x 4021 and has 8 divisors

647382 = 2 x 3 x 107897 and has 8 divisors

647383 = 11 x 229 x 257 and has 8 divisors

647384 = 2^3 x 80923 and has 8 divisors

647385 = 3 x 5 x 43159 and has 8 divisors

647386 = 2 x 89 x 3637 and has 8 divisors

647387 = 13 x 19 x 2621 and has 8 divisors

The appearance of a number with eight as a divisor is not surprising given that, out of eight consecutive numbers, one must be divisible by eight. In fact every fourth number must be divisible by 4 and it appears that in all our runs the number immediately before the first number and after the last number is divisible by 4. For example, consider the case of 647381:

647380 = 2^2 * 5 * 32369 and has 12 divisors

647381 = 7 * 23 * 4021 and has 8 divisors

647382 = 2 * 3 * 107897 and has 8 divisors

647383 = 11 * 229 * 257 and has 8 divisors

647384 = 2^3 * 80923 and has 8 divisors

647385 = 3 * 5 * 43159 and has 8 divisors

647386 = 2 * 89 * 3637 and has 8 divisors

647387 = 13 * 19 * 2621 and has 8 divisors

647388 = 2^2 * 3^2 * 7^2 * 367 and has 54 divisors

By extension, one can investigate other sorts of runs. For example, runs of numbers with the same number of prime factors.  Investigation reveals that 526095 is the only number in the range up to one million that starts off a run of 14 numbers, all with three prime factors (permalink).

526095 = 3^5 x 5 x 433 has prime factors [3, 5, 433]

526096 = 2^4 x 131 x 251 has prime factors [2, 131, 251]

526097 = 11 x 13^2 x 283 has prime factors [11, 13, 283]

526098 = 2 x 3 x 87683 has prime factors [2, 3, 87683]

526099 = 7 x 17 x 4421 has prime factors [7, 17, 4421]

526100 = 2^2 x 5^2 x 5261 has prime factors [2, 5, 5261]

526101 = 3 x 31 x 5657 has prime factors [3, 31, 5657]

526102 = 2 x 23 x 11437 has prime factors [2, 23, 11437]

526103 = 37 x 59 x 241 has prime factors [37, 59, 241]

526104 = 2^3 x 3^2 x 7307 has prime factors [2, 3, 7307]

526105 = 5 x 43 x 2447 has prime factors [5, 43, 2447]

526106 = 2 x 7 x 37579 has prime factors [2, 7, 37579]

526107 = 3 x 157 x 1117 has prime factors [3, 157, 1117]

526108 = 2^2 x 11^2 x 1087 has prime factors [2, 11, 1087]

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]