Super Sphenic Numbers

The number associated with my diurnal age today (28262) is what might be termed a "super sphenic number" as I'll explain in this post. Firstly however, its factorisation:$$28262=2 \times 13 \times 1087$$If we reverse its digits, we get the number 26282 and this number is also sphenic:$$26282 = 2 \times 17 \times 773$$Let's now concatenate the factors of 28262 in ascending order. This gives us the number 2131087 which is also sphenic:$$2131087 = 7 \times 167 \times 1823$$28262 has a sum of digits of 20 and if we add this to the original number we get the palindromic number 28282 which is also sphenic:$$28262+20=28282 = 2 \times 79 \times 179$$The number has a product of digits of 384 and if we subtract this from the original number we get 27878 which is sphenic:$$28262 - 384 = 27878 = 2 \times 53 \times 263$$If we consider only the internal digits of 28262, we get the number 826 which is also sphenic:$$2\, 826 \, 2 \rightarrow826=2 \times 7 \times 59$$When a sphenic number is considered as a sphenic brick then it has an associated number in the form of the brick's surface area. In the case of 28262, this associated surface area is 32662 square units and this number too is sphenic:$$32662 = 2 \times 7 \times 2333$$28262 has a sum of proper divisors that is also sphenic:$$ \sigma(28262) - 28262 =17434 = 2 \times 23 \times 379 $$The number has a sum of prime factors (1102) that is sphenic:$$2 +13+1087=1102 = 2 \times 19 \times 29$$28262 has a totient of 13032 which is not sphenic but its cototient (number - totient) of 15230 is:$$28262 - 13032 = 15230 = 2 \times 5 \times 1523$$So we can see that 28262 may well be termed a super sphenic number because of the above associations.

Some Interesting Integer Ratios

The number \( \textbf{27800} \) (my diurnal age today) has the following divisors: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 139, 200, 278, 556, 695, 1112, 1390, 2780, 3475, 5560, 6950, 13900, 27800. If we concatenate these divisors in the order shown (from smallest to largest) we get the rather large integer shown below.

1245810202540501001392002785566951112139027803475556069501390027800

The sum of these divisors is 65100 and it so happens that 65100 divides this concatenated number without remainder to give:

19136869470668218147342592712241952567419781927427896612924578

Numbers with this property belong to OEIS A308486: numbers such that the sum of divisors divides the concatenation (in ascending order) of divisors. The initial members up to 40000 are (permalink):

1, 2, 6, 10, 40, 98, 112, 120, 1904, 2680, 4040, 4128, 5136, 9920, 12224, 17900, 20880, 27800

Looking at the number 98 in the list we see that its divisors are 1, 2, 7, 14, 49 and 98 that have a total sum of 171 and whose concatenated divisors form the number 127144998. Thus we have:$$ \begin{align} \frac{127144998}{171} &= \frac{2 \times 3^3 \times 19 \times 123923}{3^2 \times 19} \\ &= 2 \times 3 \times 123932 \\ &=743538 \end{align} $$What if we consider the concatenation of a number's factors (with repetition) and whether it can be divided by its sum of factors (again with repetition). We need to ignore the prime numbers or else they will all get included. There are 277 composite numbers satisfying the criterion in the range up to 40000 (permalink):

8, 14, 20, 24, 27, 62, 125, 150, 160, 180, 194, 218, 300, 343, 452, 510, 512, 548, 570, 605, 612, 627, 651, 662, 663, 720, 935, 1183, 1210, 1235, 1331, 1335, 1575, 1676, 1994, 2090, 2106, 2130, 2197, 2218, 2303, 2337, 2345, 2350, 2428, 2436, 2640, 2667, 2675, 2679, 2744, 3087, 3102, 3108, 3168, 3237, 3275, 3399, 3509, 3553, 3740, 3835, 4029, 4046, 4125, 4180, 4347, 4384, 4392, 4410, 4488, 4565, 4704, 4805, 4913, 5015, 5037, 5047, 5120, 5551, 5829, 5888, 5968, 6223, 6250, 6549, 6662, 6666, 6747, 6837, 6859, 6888, 6923, 7030, 7189, 7337, 7448, 7449, 7462, 7488, 8000, 8064, 8165, 8246, 8421, 8624, 8742, 8853, 8949, 9000, 9331, 9344, 9709, 9804, 9975, 9990, 10017, 10125, 10146, 10240, 10387, 10800, 10854, 10865, 10879, 10989, 11045, 11121, 11205, 11264, 11704, 11891, 12032, 12152, 12167, 12288, 12337, 13237, 13243, 13277, 13284, 13702, 13792, 13824, 13858, 14308, 14457, 14555, 14580, 15015, 15025, 15042, 15054, 15301, 15552, 15820, 16038, 16428, 16549, 16827, 16856, 17347, 17496, 17600, 17850, 17914, 17949, 18172, 18213, 18377, 18495, 18821, 18963, 19135, 19425, 19513, 19683, 19860, 19885, 19910, 20041, 20083, 20727, 20746, 20878, 20951, 21033, 21175, 21197, 21340, 21965, 21978, 22008, 22021, 22152, 22275, 22317, 23069, 23280, 23548, 23715, 23785, 23998, 24037, 24244, 24389, 24986, 25182, 25344, 25647, 26129, 26754, 27010, 27480, 27664, 27832, 27880, 28006, 28037, 28566, 28577, 28840, 28896, 29064, 29281, 29326, 29388, 29602, 29614, 29624, 29783, 29791, 30082, 30186, 30226, 30229, 30240, 30420, 30814, 30825, 31097, 31349, 31412, 31581, 31780, 32076, 32418, 32640, 32697, 33292, 33473, 33480, 34132, 34133, 34481, 34521, 34773, 35046, 35557, 35616, 36022, 36040, 36162, 36176, 36478, 36504, 37026, 37789, 38024, 38200, 38340, 38399, 38480, 38658, 39292, 39406, 39463

This sequence of numbers is NOT to be found in the OEIS. Let's look at one of the numbers in the above list, namely 27832.$$27832=2^3 \times 7^2 \times 71$$The concatenated factors form the number 2227771 and the sum of these divisors is 91. Thus we have:$$ \begin{align} \frac{2227771}{91} &= \frac{7 \times 13 \times 24481}{7 \times 13} \\ &=24481 \end{align} $$There are other variations on the two themes covered in this post. For example, we could consider only the proper divisors of a number and look for numbers whose proper divisors, when concatenated from smallest to largest, are divisible by the sum of the proper divisors. We need to exclude prime numbers because the proper divisor in every case is 1 and thus will divide any number. There are 38 composite numbers in the range up to 40000 (permalink):

4, 15, 18, 24, 69, 208, 247, 501, 559, 565, 692, 697, 1501, 2077, 2257, 2759, 3551, 3661, 4135, 4227, 5123, 5461, 5536, 6109, 8640, 10821, 12179, 12667, 13631, 16939, 19781, 23587, 24307, 26827, 27331, 30701, 33877, 38887

Let's consider 69 in the previous list. It has proper divisors of 1, 3 and 23 that form the concatenated number 1323 with a sum of 27. Thus we have:$$ \begin{align} \frac{1323}{27} &= \frac{3^3 \times 7^2}{3^3} \\ &=7^2 \\ &=49 \end{align}$$Again this sequence is NOT to be found in the OEIS. Interestingly, the concatenated proper divisors of 8640 (one of the sequence members) form the following enormous cancatenated number:

12345689101215161820242730323640454854606472809096108120135144160180192216240270288320360432480540576720864960108014401728216028804320

Encoding Secret Numbers

The previous post about Heinz numbers got me thinking about how secret numbers that we might use as pass numbers for access to various services could be encoded. Let say we have a secret number 659347 that we want to encode using the one to one correspondence between Heinz numbers and integer sequences. Let's treat 659347 as a Heinz number, generate the corresponding integer sequence and then concatenate the numbers in the sequence:$$ \begin{align} \text{Heinz}(659347) &=(134, 19, 6) \\ &\rightarrow 134196 \end{align}$$If we concatenate the numbers in this sequence, there is ambiguity because the sequences (1341, 96) or (13419, 6) give the same result when the members are concatenated. It would thus be necessary to introduce a dot between the sequence members so as to resemble an IP address. An additional dummy 0 could be added so that it does indeed look like an IP address (0 cannot appear in any Heinz integer sequence). Thus we would have (permalink):$$ \begin{align} \text{Heinz}(659347) &= (134, 19, 6) \\ &\rightarrow 134.19.6  \text{ or} \\ &\rightarrow 134.19.6.0 \end{align}$$The point is that 134.19.6 or 134.19.6.0 can be written in plain sight and there's no way to decode the encrypted secret number (unless you know the magic word: Heinz).

An alternative method would be use the Gray Code. I've written about this in a post titled Gray Code in June of 2023. Here the secret number undergoes the following transformation:$$ \begin{align}659347 &= 10100000111110010011_2 \\ &\rightarrow 11110000100001011010_2 \\ &= 985178 \end{align} $$Again this number (985178) can be written in plain sight and there's no way to decode it without the magic phrase: Gray Code. Another encoding mechanism is Binary Complement. See my post Binary Complement from October of 2024. Here the encoding goes as follows:$$ \begin{align} 659347 &= 10100000111110010011_2\\ &\rightarrow 01011111000001101100_2\\ &=389228 \end{align} $$There is a problem once we try to decode from 389228. If we convert this number to binary then we don't have a leading zero anymore, we have 1011111000001101100. We need to add this leading zero because binary complements always have a leading zero. Once we do this, we can flip each 0 to 1 and each 1 to 0 to recover our secret number once we convert from binary. If one wanted to be really devious then two or even three encoding techniques could be used sequentially. 

Other Special Classes of Interprimes

On the 1st November 2023, I posted on A Special Class of Interprime and these were non-palindromic composite numbers located between twin primes which, when reversed, are also located between twin primes. Some work both ways while some are only one way because they end in a zero. Figure 1 shows an example of the former while Figure 2 shows an example of the latter.

Figure 1

Figure 2

Today I turned 27738 days old and this number is an interprime number between twin primes which when concatenated with itself forms a number which is also an interprime between twin primes. The result for 27738 is shown in Figure 3.

Figure 3

Numbers of this sort belong to OEIS A235109 :

A235109     Averages q of twin prime pairs, such that q concatenated to q is also the average of a twin prime pair.

The initial members are (permalink):

42, 102, 108, 180, 192, 270, 312, 420, 522, 660, 822, 882, 1230, 1482, 4242, 4788, 8820, 10332, 11550, 13692, 14550, 14562, 14868, 15732, 17910, 18522, 20550, 21648, 22620, 23670, 23832, 26262, 27738, 35838, 38922, 39042, 40128, 42018, 43962, 44532, 46440

As a variation on this, we could concatenate an interprime with its reversal, thus forming a palindrome. This is shown in Figure 4.

Figure 4

Up to 40000, the initial interprimes with this property are (permalink) 240, 270, 2142, 8388, 22092, 22962, 23832, 24420, 24918, 26262, 27690 and 28110. The sequence does not appear in the OEIS. The members of this sequence are, to be fair, rather sparse and could be made more numerous if the condition that the interprime lay between twin primes was relaxed. If we simply require that the interprime, when concatenated with its reverse, is also an interprime then in the range up to 40000, 263 numbers satisfy. The numbers are (permalink):

9, 15, 21, 42, 93, 102, 105, 108, 160, 240, 246, 270, 279, 324, 386, 432, 754, 810, 909, 933, 1092, 1302, 1452, 1611, 1998, 2142, 2205, 2295, 2322, 2336, 2470, 2568, 2667, 2892, 2900, 2946, 3021, 3326, 3423, 3453, 3465, 3558, 3588, 3627, 3672, 3736, 3885, 3921, 4002, 4065, 4076, 4131, 4353, 4422, 4646, 4742, 4785, 5193, 5439, 5481, 5502, 5529, 5607, 5804, 6107, 6340, 6376, 6798, 6969, 7182, 7212, 7494, 8097, 8169, 8388, 8437, 8844, 8908, 8985, 9394, 9678, 9865, 10008, 10101, 10794, 10815, 10875, 10944, 10998, 11226, 11445, 11523, 11817, 12024, 12111, 12252, 12489, 12500, 12514, 12826, 12947, 13056, 13101, 13320, 13374, 13482, 13560, 13674, 13740, 13881, 13965, 14064, 14415, 14592, 14715, 15015, 15087, 15534, 15664, 16230, 16396, 16799, 17388, 17529, 17958, 18042, 18288, 18360, 18447, 18531, 18737, 19149, 19314, 19548, 19704, 19857, 20022, 20049, 20057, 20225, 20358, 20403, 20687, 20745, 20751, 20808, 21015, 21104, 21189, 21202, 21381, 21404, 21558, 21969, 22092, 22272, 22719, 22866, 22904, 22962, 23124, 23631, 23832, 24036, 24144, 24333, 24420, 24522, 24804, 24855, 24918, 25001, 25080, 25305, 25417, 25455, 25470, 25578, 25595, 25761, 25932, 25960, 26180, 26262, 26412, 26582, 26637, 26675, 26748, 27075, 27429, 27546, 27597, 27690, 27999, 28110, 28117, 28253, 28314, 28410, 28629, 28692, 28869, 29247, 29577, 29720, 29826, 29865, 29937, 30106, 30165, 30217, 30270, 30693, 31149, 31152, 31182, 31269, 31536, 31617, 31653, 31977, 32244, 32325, 32700, 32914, 33186, 33288, 33573, 33588, 33621, 33639, 33854, 33939, 34125, 34290, 34412, 34590, 34683, 34743, 34874, 34962, 35094, 35421, 35674, 35802, 36003, 36442, 36486, 36648, 36694, 36764, 37220, 37514, 37548, 38385, 38856, 39093, 39159, 39447, 39627, 39852, 39999

Let's take 93 from the previous list as an example. It is an interprime that lies midway between 89 and 97. Concatenated with its reverse (39), we get 9339 and this number is midway between 9337 and 9341. Figure 5 illustrates this.

Figure 5

Similarly we could relax the interprime condition for interprimes that are concatenated with themselves (but not reversed). There are 345 interprimes in the range up to 40000 that qualify. An example is 21, an interprime between 19 and 23, that forms 2121, an interprime between 2113 and 2129.

Figure 6

The interprimes between 27700 and 40000 with this property are (permalink):

..., 27738, 27888, 27945, 27990, 28281, 28515, 28613, 28740, 28815, 28851, 28994, 29013, 29170, 29237, 29307, 29448, 29835, 29953, 30000, 30038, 30264, 30300, 30310, 30378, 30468, 30555, 30846, 30856, 30902, 31080, 31122, 31269, 31335, 31347, 31660, 31854, 31960, 32298, 32361, 32715, 32925, 32990, 33018, 33235, 33351, 33465, 33594, 33717, 33840, 33860, 34224, 34734, 34743, 34848, 35325, 35556, 35571, 35838, 35980, 36189, 36462, 36680, 37008, 37053, 37176, 37576, 37850, 38076, 38238, 38310, 38331, 38685, 38922, 39042, 39084, 39093, 39105, 39363, 39378, 39447, 39546, 39691, 39765, 39774, 39894, ...

Lastly, if we relax the interprime condition that the interprime and its reverse must lie between twin primes, then there are 629 numbers that satisfy in the range up to 40000 (permalink) but I won't list those here.

Biprime Prime Time

For some reason, it only just occurred to me that I can search this blog for the occurrence of particular numbers. Usually I search the OEIS first and if nothing of interest comes up, I search my Bespoken for Sequences database. If there's nothing there, I'll search the airtable.com database and if nothing turns up, I'll search for the number in the OEIS b-files. However, searching this blog should probably be my second priority if nothing comes up in the OEIS. After all I have hundreds of posts and thousands of numbers.

The theme of this post concerns biprimes or semiprimes where there is a connection between one prime and the other. The idea derived from the number associated with my diurnal age today: 27149. This number is a member of OEIS A045925:

A045925

a(\(n\)) = \(n\) * Fibonacci(\(n\)).

When \(n=17\), we have Fibonacci(17) = 1597 which is prime and 27149 = 17 x 1597. Very few of the members of this sequence are biprimes because \(n\) can be composite and\or Fibonacci(\(n\)) can be composite. The biprimes are listed below (in bold):

3 --> 6 = 2 * 3
5 --> 25 = 5^2
7 --> 91 = 7 * 13
11 --> 979 = 11 * 89
13 --> 3029 = 13 * 233
17 --> 27149 = 17 * 1597
23 --> 659111 = 23 * 28657
29 --> 14912641 = 29 * 514229
43 --> 18640260791 = 43 * 433494437
47 --> 139647108431 = 47 * 2971215073

We are looking for biprimes of the form prime multiplied by some function of that prime where the function generates a new prime. An easy source of such numbers arises from OEIS A073065: 

A073065

a(\(n\)) = prime(\(n\)) * prime(prime(\(n\))).

The sequence begins: 6, 15, 55, 119, 341, 533, 1003, 1273, 1909, 3161, 3937, 5809, 7339, 8213, 9917, 12773, 16343, 17263, 22177, 25063, 26791, 31679, 35773, 41029

The breakdown is as follows:

1 --> 2 x 3 = 6

2 --> 3 x 5 = 15

3 --> 5 x 11 = 55

4 --> 7 x 17 = 119

5 --> 11 x 31 = 341

6 --> 13 x 41 = 533

7 --> 17 x 59 = 1003

8 --> 19 x 67 = 1273

9 --> 23 x 83 = 1909

10 --> 29 x 109 = 3161

11 --> 31 x 127 = 3937

12 --> 37 x 157 = 5809

13 --> 41 x 179 = 7339

14 --> 43 x 191 = 8213

15 --> 47 x 211 = 9917

16 --> 53 x 241 = 12773

17 --> 59 x 277 = 16343

18 --> 61 x 283 = 17263

19 --> 67 x 331 = 22177

20--> 71 x 353 = 25063

21 --> 73 x 367 = 26791

22 --> 79 x 401 = 31679

23 --> 83 x 431 = 35773

24 --> 89 x 461 = 41029

25 --> 97 x 509 = 49373

Another way to find "special" biprimes is to determine which of them concatenate to form new primes, either by smaller with larger or larger with smaller. Such biprimes form OEIS A330441:

A330441

Semiprimes \(p \times q\) such that the concatenations of \(p\) and \(q\) in both orders are prime.

Here is a list of the initial members (508) of this sequence up to 40000:

21, 33, 51, 93, 111, 133, 177, 201, 219, 247, 253, 327, 411, 427, 573, 589, 679, 687, 763, 793, 813, 889, 993, 1077, 1081, 1119, 1243, 1339, 1347, 1401, 1411, 1497, 1501, 1603, 1623, 1651, 1671, 1821, 1839, 1843, 1851, 1981, 2019, 2047, 2059, 2103, 2157, 2199, 2217, 2469, 2479, 2629, 2761, 2787, 2841, 2923, 3031, 3039, 3057, 3097, 3099, 3133, 3153, 3409, 3439, 3543, 3579, 3661, 3711, 3787, 3829, 3883, 3973, 4063, 4171, 4303, 4309, 4369, 4381, 4429, 4443, 4593, 4681, 4711, 4771, 4821, 4837, 4843, 4881, 4971, 4989, 5001, 5097, 5191, 5299, 5533, 5611, 5721, 5761, 5803, 5971, 5989, 6181, 6207, 6429, 6457, 6511, 6559, 6613, 6639, 6799, 6891, 7003, 7131, 7143, 7153, 7233, 7323, 7327, 7363, 7501, 7509, 7633, 7711, 7737, 8121, 8173, 8197, 8227, 8347, 8367, 8529, 8659, 8743, 8751, 8871, 9147, 9211, 9217, 9229, 9307, 9313, 9357, 9493, 9523, 9589, 9793, 9853, 9903, 10063, 10171, 10249, 10297, 10351, 10383, 10407, 10441, 10483, 10519, 10537, 10777, 10807, 10843, 10849, 10911, 10963, 11107, 11307, 11539, 11581, 11623, 11629, 11653, 11707, 11733, 11769, 11793, 11797, 11851, 12001, 12057, 12133, 12187, 12193, 12477, 12643, 12759, 12961, 13027, 13117, 13153, 13387, 13471, 13749, 13771, 13813, 13951, 13993, 14187, 14257, 14623, 14677, 14757, 14803, 14977, 15049, 15127, 15151, 15153, 15177, 15229, 15247, 15297, 15513, 15571, 15697, 15769, 15843, 15883, 15969, 16003, 16009, 16143, 16257, 16347, 16357, 16387, 16501, 16507, 16521, 16543, 16563, 16593, 16719, 16837, 17089, 17131, 17173, 17269, 17371, 17403, 17503, 17517, 17521, 17607, 17769, 17779, 17803, 17833, 17953, 18111, 18219, 18247, 18319, 18673, 18697, 18709, 18789, 18829, 18831, 18841, 18871, 18943, 18949, 19021, 19033, 19039, 19059, 19353, 19357, 19407, 19587, 19627, 19689, 19693, 19713, 19741, 19797, 19807, 19897, 19911, 19939, 19959, 20059, 20191, 20221, 20317, 20379, 20581, 20613, 20671, 20697, 20701, 20841, 21117, 21133, 21151, 21171, 21253, 21439, 21457, 21477, 21691, 21703, 21759, 22069, 22107, 22207, 22267, 22459, 22507, 22759, 22773, 22947, 23073, 23317, 23377, 23503, 23527, 23713, 23731, 23889, 24031, 24193, 24313, 24343, 24487, 24607, 24613, 24643, 24657, 24711, 24721, 24949, 25081, 25131, 25267, 25291, 25293, 25507, 25549, 25651, 25729, 25777, 25813, 25887, 25963, 26007, 26077, 26089, 26097, 26139, 26167, 26241, 26329, 26349, 26359, 26401, 26629, 26761, 26989, 27049, 27309, 27469, 27471, 27493, 27543, 27619, 27723, 27781, 28117, 28129, 28189, 28383, 28417, 28563, 28831, 28869, 28993, 29113, 29227, 29239, 29407, 29479, 29533, 29571, 29647, 29661, 29703, 29707, 29713, 29731, 30273, 30333, 30499, 30507, 30511, 30607, 30669, 30721, 30729, 30799, 30979, 30999, 31071, 31087, 31273, 31279, 31363, 31377, 31453, 31459, 31483, 31549, 31621, 31701, 31711, 31881, 31969, 32167, 32197, 32281, 32313, 32667, 32743, 32961, 33043, 33081, 33103, 33519, 33531, 33591, 33913, 34063, 34117, 34179, 34321, 34339, 34341, 34417, 34459, 34609, 34653, 34789, 34813, 34933, 34957, 35173, 35299, 35359, 35383, 35463, 35481, 35601, 35691, 35743, 35779, 35943, 35953, 36019, 36181, 36213, 36259, 36303, 36367, 36601, 36609, 36649, 36667, 36759, 36763, 36769, 36807, 36961, 37029, 37063, 37099, 37239, 37267, 37351, 37353, 37399, 37621, 37803, 38059, 38109, 38131, 38137, 38347, 38359, 38389, 38523, 38623, 38647, 38683, 38697, 38929, 38937, 39007, 39049, 39427, 39487, 39553, 39613, 39649, 39823, 39931, 39967

 For example, 27309= 3 * 9103 and the concatenations 39103 and 91033 are both prime.

Concatenations of Squares and Cubes

I was surprised that the number associated with my diurnal age today, 27125, didn't show up in the OEIS as a concatenation of two cubes, namely \(3^3\) and \(5^3\). This prompted me to list all numbers of the form \(n^3 \, | \, m^3\) where \(n\) and \(m\) are integers (not necessarly distinct). Here is a permalink to the SageMath code that generates the list up to 40,000 and here is the list:

11, 18, 81, 88, 127, 164, 271, 278, 641, 648, 827, 864, 1125, 1216, 1251, 1258, 1343, 1512, 1729, 2161, 2168, 2727, 2764, 3431, 3438, 5121, 5128, 6427, 6464, 7291, 7298, 8125, 8216, 8343, 8512, 8729, 10001, 10008, 11000, 11331, 11728, 12197, 12527, 12564, 12744, 13311, 13318, 13375, 14096, 14913, 15832, 16859, 17281, 17288, 18000, 19261, 21627, 21664, 21971, 21978, 27125, 27216, 27343, 27441, 27448, 27512, 27729, 33751, 33758, 34327, 34364

Some are more difficult to spot than others. What about concatenations of square numbers? Here is a permalink to the SageMath code that generates a list of numbers up to 40,000 and here is the list of numbers of the form  \(n^2 \, | \, m^2\) where \(n\) and \(m\) are integers (not necessarly distinct):

11, 14, 19, 41, 44, 49, 91, 94, 99, 116, 125, 136, 149, 161, 164, 169, 181, 251, 254, 259, 361, 364, 369, 416, 425, 436, 449, 464, 481, 491, 494, 499, 641, 644, 649, 811, 814, 819, 916, 925, 936, 949, 964, 981, 1001, 1004, 1009, 1100, 1121, 1144, 1169, 1196, 1211, 1214, 1219, 1225, 1256, 1289, 1324, 1361, 1400, 1441, 1444, 1449, 1484, 1529, 1576, 1616, 1625, 1636, 1649, 1664, 1676, 1681, 1691, 1694, 1699, 1729, 1784, 1841, 1900, 1961, 1964, 1969, 2251, 2254, 2259, 2516, 2525, 2536, 2549, 2561, 2564, 2569, 2581, 2891, 2894, 2899, 3241, 3244, 3249, 3611, 3614, 3616, 3619, 3625, 3636, 3649, 3664, 3681, 4001, 4004, 4009, 4100, 4121, 4144, 4169, 4196, 4225, 4256, 4289, 4324, 4361, 4400, 4411, 4414, 4419, 4441, 4484, 4529, 4576, 4625, 4676, 4729, 4784, 4841, 4844, 4849, 4900, 4916, 4925, 4936, 4949, 4961, 4964, 4981, 5291, 5294, 5299, 5761, 5764, 5769, 6251, 6254, 6259, 6416, 6425, 6436, 6449, 6464, 6481, 6761, 6764, 6769, 7291, 7294, 7299, 7841, 7844, 7849, 8116, 8125, 8136, 8149, 8164, 8181, 8411, 8414, 8419, 9001, 9004, 9009, 9100, 9121, 9144, 9169, 9196, 9225, 9256, 9289, 9324, 9361, 9400, 9441, 9484, 9529, 9576, 9611, 9614, 9619, 9625, 9676, 9729, 9784, 9841, 9900, 9961, 10016, 10025, 10036, 10049, 10064, 10081, 10241, 10244, 10249, 10891, 10894, 10899, 11024, 11089, 11156, 11225, 11296, 11369, 11444, 11521, 11561, 11564, 11569, 11600, 11681, 11764, 11849, 11936, 12025, 12116, 12125, 12136, 12149, 12164, 12181, 12209, 12251, 12254, 12259, 12304, 12401, 12500, 12601, 12704, 12809, 12916, 12961, 12964, 12969, 13025, 13136, 13249, 13364, 13481, 13600, 13691, 13694, 13699, 13721, 13844, 13969, 14096, 14225, 14356, 14416, 14425, 14436, 14441, 14444, 14449, 14464, 14481, 14489, 14624, 14761, 14900, 15041, 15184, 15211, 15214, 15219, 15329, 15476, 15625, 15776, 15929, 16001, 16004, 16009, 16084, 16100, 16121, 16144, 16169, 16196, 16225, 16241, 16256, 16289, 16324, 16361, 16400, 16441, 16484, 16529, 16561, 16576, 16625, 16676, 16724, 16729, 16784, 16811, 16814, 16819, 16841, 16889, 16900, 16916, 16925, 16936, 16949, 16961, 16964, 16981, 17056, 17225, 17396, 17569, 17641, 17644, 17649, 17744, 17921, 18100, 18281, 18464, 18491, 18494, 18499, 18649, 18836, 19025, 19216, 19361, 19364, 19369, 19409, 19604, 19616, 19625, 19636, 19649, 19664, 19681, 19801, 20251, 20254, 20259, 21161, 21164, 21169, 22091, 22094, 22099, 22516, 22525, 22536, 22549, 22564, 22581, 23041, 23044, 23049, 24011, 24014, 24019, 25001, 25004, 25009, 25100, 25121, 25144, 25169, 25196, 25225, 25256, 25289, 25324, 25361, 25400, 25441, 25484, 25529, 25576, 25616, 25625, 25636, 25649, 25664, 25676, 25681, 25729, 25784, 25841, 25900, 25961, 26011, 26014, 26019, 27041, 27044, 27049, 28091, 28094, 28099, 28916, 28925, 28936, 28949, 28964, 28981, 29161, 29164, 29169, 30251, 30254, 30259, 31361, 31364, 31369, 32416, 32425, 32436, 32449, 32464, 32481, 32491, 32494, 32499, 33641, 33644, 33649, 34811, 34814, 34819, 36001, 36004, 36009, 36100, 36116, 36121, 36125, 36136, 36144, 36149, 36164, 36169, 36181, 36196, 36225, 36256, 36289, 36324, 36361, 36400, 36441, 36484, 36529, 36576, 36625, 36676, 36729, 36784, 36841, 36900, 36961, 37211, 37214, 37219, 38441, 38444, 38449, 39691, 39694, 39699

We can thin the above list of numbers by requiring that the number formed by the concatenation be a square number (permalink):

49, 169, 361, 1225, 1444, 1681, 3249, 4225, 4900, 15625, 16900, 36100

Here we see that \(36100 = 6^2 \, | \, 10^2 = 190^2 \).

While we're at it, let's consider concatenations of fourth powers. Here is a list (permalink) of numbers of the form \(n^4 \, | \, m^4\) where \(n\) and \(m\) are integers (not necessarly distinct):

11, 116, 161, 181, 811, 1256, 1616, 1625, 1681, 2561, 6251, 8116, 8181, 11296, 12401, 12961, 14096, 16256, 16561, 16625, 24011, 25616, 25681

We don't have to limit ourselves to concatenations of pairs of powers. We can concatenate three powers as easily as two. Let's consider numbers that are a concatenation of three square numbers (permalink):

111, 114, 119, 141, 144, 149, 191, 194, 199, 411, 414, 419, 441, 444, 449, 491, 494, 499, 911, 914, 919, 941, 944, 949, 991, 994, 999, 1116, 1125, 1136, 1149, 1161, 1164, 1169, 1181, 1251, 1254, 1259, 1361, 1364, 1369, 1416, 1425, 1436, 1449, 1464, 1481, 1491, 1494, 1499, 1611, 1614, 1619, 1641, 1644, 1649, 1691, 1694, 1699, 1811, 1814, 1819, 1916, 1925, 1936, 1949, 1964, 1981, 2511, 2514, 2519, 2541, 2544, 2549, 2591, 2594, 2599, 3611, 3614, 3619, 3641, 3644, 3649, 3691, 3694, 3699, 4116, 4125, 4136, 4149, 4161, 4164, 4169, 4181, 4251, 4254, 4259, 4361, 4364, 4369, 4416, 4425, 4436, 4449, 4464, 4481, 4491, 4494, 4499, 4641, 4644, 4649, 4811, 4814, 4819, 4911, 4914, 4916, 4919, 4925, 4936, 4941, 4944, 4949, 4964, 4981, 4991, 4994, 4999, 6411, 6414, 6419, 6441, 6444, 6449, 6491, 6494, 6499, 8111, 8114, 8119, 8141, 8144, 8149, 8191, 8194, 8199, 9116, 9125, 9136, 9149, 9161, 9164, 9169, 9181, 9251, 9254, 9259, 9361, 9364, 9369, 9416, 9425, 9436, 9449, 9464, 9481, 9491, 9494, 9499, 9641, 9644, 9649, 9811, 9814, 9819, 9916, 9925, 9936, 9949, 9964, 9981, 10011, 10014, 10019, 10041, 10044, 10049, 10091, 10094, 10099, 11001, 11004, 11009, 11100, 11121, 11144, 11169, 11196, 11211, 11214, 11219, 11225, 11256, 11289, 11324, 11361, 11400, 11441, 11444, 11449, 11484, 11529, 11576, 11616, 11625, 11636, 11649, 11664, 11676, 11681, 11691, 11694, 11699, 11729, 11784, 11841, 11900, 11961, 11964, 11969, 12111, 12114, 12119, 12141, 12144, 12149, 12191, 12194, 12199, 12251, 12254, 12259, 12516, 12525, 12536, 12549, 12561, 12564, 12569, 12581, 12891, 12894, 12899, 13241, 13244, 13249, 13611, 13614, 13616, 13619, 13625, 13636, 13649, 13664, 13681, 14001, 14004, 14009, 14100, 14121, 14144, 14169, 14196, 14225, 14256, 14289, 14324, 14361, 14400, 14411, 14414, 14419, 14441, 14444, 14449, 14484, 14491, 14494, 14499, 14529, 14576, 14625, 14676, 14729, 14784, 14841, 14844, 14849, 14900, 14916, 14925, 14936, 14949, 14961, 14964, 14981, 15291, 15294, 15299, 15761, 15764, 15769, 16116, 16125, 16136, 16149, 16161, 16164, 16169, 16181, 16251, 16254, 16259, 16361, 16364, 16369, 16416, 16425, 16436, 16449, 16464, 16481, 16491, 16494, 16499, 16641, 16644, 16649, 16761, 16764, 16769, 16811, 16814, 16819, 16911, 16914, 16916, 16919, 16925, 16936, 16941, 16944, 16949, 16964, 16981, 16991, 16994, 16999, 17291, 17294, 17299, 17841, 17844, 17849, 18116, 18125, 18136, 18149, 18164, 18181, 18411, 18414, 18419, 19001, 19004, 19009, 19100, 19121, 19144, 19169, 19196, 19225, 19256, 19289, 19324, 19361, 19400, 19441, 19484, 19529, 19576, 19611, 19614, 19619, 19625, 19641, 19644, 19649, 19676, 19691, 19694, 19699, 19729, 19784, 19841, 19900, 19961, 22511, 22514, 22519, 22541, 22544, 22549, 22591, 22594, 22599, 25116, 25125, 25136, 25149, 25161, 25164, 25169, 25181, 25251, 25254, 25259, 25361, 25364, 25369, 25416, 25425, 25436, 25449, 25464, 25481, 25491, 25494, 25499, 25611, 25614, 25619, 25641, 25644, 25649, 25691, 25694, 25699, 25811, 25814, 25819, 25916, 25925, 25936, 25949, 25964, 25981, 28911, 28914, 28919, 28941, 28944, 28949, 28991, 28994, 28999, 32411, 32414, 32419, 32441, 32444, 32449, 32491, 32494, 32499, 36111, 36114, 36116, 36119, 36125, 36136, 36141, 36144, 36149, 36161, 36164, 36169, 36181, 36191, 36194, 36199, 36251, 36254, 36259, 36361, 36364, 36369, 36416, 36425, 36436, 36449, 36464, 36481, 36491, 36494, 36499, 36641, 36644, 36649, 36811, 36814, 36819, 36916, 36925, 36936, 36949, 36964, 36981

Here we see that \(36981= 6^2 \, | \, 3^2 \, | \,9^2\). Once again, we can thin the above numbers by adding the requirement that the number formed by the concatenation be a square number. In this case, we get (permalink):

144, 441, 1369, 1936, 11449, 11664, 14400, 16641, 36481

Here we see that \(36481=6^2 \, | \, 2^2 \, | \,9^2 = 191^2\). None of these sequences of numbers appear in the OEIS as far as I'm aware and I certainly won't be adding them. So nothing of deep mathematical significance in this post, just playing around with powers of numbers and concatenating them. Of course, I've written about Primes Formed By Concatenation quite recently on June 17th 2023.

Sums and Concatenations of Cubes and Squares

There's something very obvious about the number associated with my diurnal age today. The number is 27064 and the cubes (27 and 64) stand out clearly. In fact 27064 can be written as a sum of two cubes:$$ \begin{align} 27064 &=27000+64\\&=30^3+4^3 \end{align}$$Unfortunately, the number cannot be written as a concatenation of two cubes because the zero gets in the way. The problem is that 4 cubed has only two digits. However, the cubes of the numbers from 5 to 9 all have three digits and so the zero disappears. This allows us to write the following numbers as both sums and concatenations of two cubes. The symbol | indicates concatenation$$ \begin{align} 27125 =30^3+5^3 = 3^3|5^3\\27216 =30^3+6^3 = 3^3|6^3\\27343 = 30^3+7^3 = 3^3|7^3\\27512 = 30^3+ 8^3 = 3^3|8^3\\27729 = 30^3+9^3=3^3|9^3 \end{align} $$This series of numbers is the last that will occur in my lifetime because the next such sets of numbers will begin with 64125. However, if we were to consider sums of squares and concatenations of squares then I may see these come to pass. Consider the following sets of numbers, some of which occur more than once (permalink).$$ \begin{align} 36100= 114^2+152^2=6^2|10^2\\36121 =20^2+ 189^2=6^2|11^2\\36121 =61^2+ 180^2=6^2|11^2\\36196=40^2+ 186^2=6^2|14^2\\36324 =90^2+ 168^2=6^2|18^2\\36361 =60^2 +181^2=6^2|19^2\\36361=125^2+ 144^2=6^2|19^2\\36441=96^2+ 165^2=6^2|21^2\\36529=48^2+ 185^2=6^2|23^2\\36625=12^2+ 191^2=6^2|25^2\\36625=56^2+ 183^2=6^2|25^2\\36625=65^2+ 180^2=6^2|25^2\\36625=105^2+ 160^2=6^2|25^2\\36676=24^2+190^2=6^2|26^2\\36676=80^2+174^2=6^2|26^2\\36900 =6^2+ 192^2=6^2|30^2\\36900=48^2+ 186^2=6^2|30^2\\36900= 120^2+ 150^2=6^2|30^2 \end{align} $$The first of these numbers (36100) corresponds to Monday, February 3rd, 2048 by which time I'll be almost 88. Maybe I'll make it, maybe I won't.

What's Special About 45162?

I want to start a series of posts that highlight special numbers. Mostly I'll be drawing on material from earlier posts where a number was mentioned but perhaps not given the prominence that it deserved. Today's number is 45162 and it's unique in the number range from one to one million. Why?

\(45162\)

The reason is by no means obvious but it has to do with the catcatenation of its prime factors which are:

 \(2 \times 3^2 \times 13 \times 193\)

When we concatenate these prime factors, we get the following number:

\(23313193\)

This number happens to be a prime. In itself, this doesn't make 45162 very special, let alone unique. About 13.7% of numbers in the range up to one million have the property that, when their prime factors are concatenated in order from lowest to highest, the resulting concatenated number is prime. Such a prime is called the home prime of that number and thus 23313193 is the home prime of 45162.

What makes 45162 unique is that it is the start of a chain of eight consecutive numbers that all have this property as shown in Figure 1:

Figure 1

Quite remarkable and it's no wonder that this octet is the only one in the range up to one million. Interestingly, none of the home primes associated with this octet of numbers contains the digit 8. I first mentioned these numbers in a post titled One Step Away on January 31st 2022. 45162 is given prominence because it starts off the sequence.

26262: A Special Palindrome

Indeed, "There are things that drift away, like our endless numbered days" and 26262 is one of them. Today I turned 26262 days old and this number should pass away perhaps only after it has received its due attention. 

So what makes this particular palindrome special? Well, to begin with, it's sandwiched between two primes and that doesn't happen often as OEIS A113838 reveals.