What's Special About 7658?

I can thank Cliff Pickover a tweet for identifying what's special about the number 7658. See Figure 1 which is a screenshot of his tweet.

Figure 1: link

So 7658 is the largest number with distinct digits that doesn't have any digits in common with its cube. 

\(7658^3 =     449103134312\)

Here is a table of all numbers with the property that there are no digits in common (permalink).

number   cube

  2        8

  3        27

  7        343

  8        512

  27       19683

  43       79507

  47       103823

  48       110592

  52       140608

  53       148877

  63       250047

  68       314432

  92       778688

  157      3869893

  172      5088448

  187      6539203

  192      7077888

  263      18191447

  378      54010152

  408      67917312

  423      75686967

  458      96071912

  468      102503232

  478      109215352

  487      115501303

  527      146363183

  587      202262003

  608      224755712

  648      272097792

  692      331373888

  823      557441767

  843      599077107

  918      773620632

  1457     3092990993

  1587     3996969003

  1592     4034866688

  4657     100999381393

  4732     105958111168

  5692     184414333888

  6058     222324747112

  6378     259449922152

  7658     449103134312

This fact got me interested in finding out the largest number with distinct digits that has no digits in common with its square. Checking in the range up to one million, which is about the limit for the online SageMathCell, I found the number to be 639172 whose square is    408540845584 (permalink). I strongly suspect that this is the largest number. So we have:

\(639172^2 =   408540845584\)

What about fourth powers? What is the largest number that, when raised to the fourth power, has no digits in common with the base number? That number turns out to be 2673.

\(2673^4 = 51050010415041\)

There doesn't appear to be any numbers satisfying the fifth power but 92 is the largest such number when sixth powers are involved:

\(92^6 =  606355001344\)

I'll leave off there. So, in summary, our investigation into what is special about 7658 led us to discover some associated numbers (92, 2673 and 639172) that are the largest possible numbers when powers of 6, 4 and 2 are considered.

What's Special About 26840?

Among all five digit numbers, the number 26840 has the very obvious but relatively rare property that it contains all the even digits 0, 2, 4, 6 and 8. It's so obvious that it's easy to miss. I'd noticed the property earlier in the day when I was analysing the number because it represents my diurnal age and didn't give it much thought. It was later in the day that it struck me that such an occurrence is not all that common in the span of all five digit numbers from 10000 to 99999. 

In fact there are only 96 such numbers because the first digit cannot be zero so there are only four possible digits to place there. Having put one of the even digits in the first position, the second position can be filled in four ways, the next in three etc. Thus we 4 x 4 x 3 x 2 x 1 = 96. Here are the numbers in order from lowest to highest:

20468, 20486, 20648, 20684, 20846, 20864, 24068, 24086, 24608, 24680, 24806, 24860, 26048, 26084, 26408, 26480, 26804, 26840, 28046, 28064, 28406, 28460, 28604, 28640, 40268, 40286, 40628, 40682, 40826, 40862, 42068, 42086, 42608, 42680, 42806, 42860, 46028, 46082, 46208, 46280, 46802, 46820, 48026, 48062, 48206, 48260, 48602, 48620, 60248, 60284, 60428, 60482, 60824, 60842, 62048, 62084, 62408, 62480, 62804, 62840, 64028, 64082, 64208, 64280, 64802, 64820, 68024, 68042, 68204, 68240, 68402, 68420, 80246, 80264, 80426, 80462, 80624, 80642, 82046, 82064, 82406, 82460, 82604, 82640, 84026, 84062, 84206, 84260, 84602, 84620, 86024, 86042, 86204, 86240, 86402, 86420

I'll only see another six such numbers in my lifetime: 28046, 28064, 28406, 28460, 28604, 28640. Here is a permalink to the SageMath algorithm that I used to generate these numbers. This got me thinking about the 120 five digit numbers that contain only the odd digits: 1, 3, 5, 7 and 9. The algorithm is easily modified to generate these numbers (permalink). Here are the numbers:

13579, 13597, 13759, 13795, 13957, 13975, 15379, 15397, 15739, 15793, 15937, 15973, 17359, 17395, 17539, 17593, 17935, 17953, 19357, 19375, 19537, 19573, 19735, 19753, 31579, 31597, 31759, 31795, 31957, 31975, 35179, 35197, 35719, 35791, 35917, 35971, 37159, 37195, 37519, 37591, 37915, 37951, 39157, 39175, 39517, 39571, 39715, 39751, 51379, 51397, 51739, 51793, 51937, 51973, 53179, 53197, 53719, 53791, 53917, 53971, 57139, 57193, 57319, 57391, 57913, 57931, 59137, 59173, 59317, 59371, 59713, 59731, 71359, 71395, 71539, 71593, 71935, 71953, 73159, 73195, 73519, 73591, 73915, 73951, 75139, 75193, 75319, 75391, 75913, 75931, 79135, 79153, 79315, 79351, 79513, 79531, 91357, 91375, 91537, 91573, 91735, 91753, 93157, 93175, 93517, 93571, 93715, 93751, 95137, 95173, 95317, 95371, 95713, 95731, 97135, 97153, 97315, 97351, 97513, 97531

So getting back to 26840, it's special because it contains all the even digits exactly once. There are only 96 such numbers and so that makes this number rather special.

The 4k+1 versus 4k+3 Composite Race

It was only recently that I posted about the \(4k-1\) versus \(4k+1\) Prime Race in connection with my diurnal age of 26833 and today, aged 26835 days, I find myself involved in another race, this time the \(4k+1\) versus \(4k+3\) Composite Race. Of course \(4k+3\) is the same as \(4k-1\) but I'll stick with the former as that's the way the OEIS entry is described:

A093180

Odd composites (including 1 in the count) where the number 1 mod 4 equals the number 3 mod 4.

The sequence runs:

26835, 26851, 26855, 26859, 26867, 26871, 26875, 26883, 26887, 26895, 26899, 26923, 616771, 616775, 616779, 616795, 616831, 616835, 616839, 616847, 616853, 616857, 616861, 616865, 616869, 616875, 616881, 616885, 616889, 616893, 617013, 617017, 617021, 617025, 617031, 617035, 617259, 617263, 617267, 617329, 617361, 617367, 617373, 617377, 617381, 617385, 617391, 617395, 617399, 617405, 617409, 617415, 617419, 617423, 617427, 617433, 617437, 617441, 617445, 617451, 617457, 617461, 617465, 617475, 617511, 617515, 617519, 617525, 617529, 617535, 617541, 617545, 617549, 617553, 617557, 617561, 617565, 617569, 617573, 617577, 617583, 617683, 617687, 617701, 617705, 617711, 617715, 618523, 618527, 618531, 618535, 618539, 618543, 618595, 618599, 618603, 618607, 618611, 618615, 618639, 622795, 622799, 622803, 622807, 622811, 622817, 622821, 622825, 622829, 622833, 622837, 622841, 622845, 622989, 622993, 623005, 623109, 623113, 623117, 623121, 623125, 623129, 623133, 623137, 623141, 623145, 623149, 623153, 623157, 623161, 623165, 623169, 623175, 623179, 623183, 623187, 623191, 623195, 623199, 623203, 623207, 623213, 623217, 623301, 623307, 623311, 623315, 623319, 623325, 623331, 623335, 623339, 623345, 623349, 623357, 623361, 623365, 623369, 623373, 623377, 623381, 623395, 623399, 623405, 623409, 623413, 623425, 623429, 623435, 623441, 623445, 623449, 623453, 623457, 623461, 623465, 623469, 623473, 623593, 623597, 623601, 623605, 623609, 623613, 623805, 623809, 623813, 623817, 623821, 623825, 623829, 623833, 623837, 623843, 623847, 623895, 623899, 623903, 623907, 623911, 623915, 623919, 623931, 623937, 623941, 623945, 623951, 623955, 623961, 623967, 623971, 623975, 623981, 623987, 623993, 623997, 624001, 624005, 624011, 624015, 624019, 624023, 624027, 624039, 624043, 624051, 624055, 624059, 624063, 624091, 624095, 624101, 624105, 624109, 624113, 624117, 624123, 624127, 624131, 624137, 624143, 624147, 624153, 624157, 624161, 624167, 624171, 624175, 624179, 624183, 624187, 624235, 624239, 624245, 624249, 624255, 626923, 626927, 626933, 626937, 626941, 626945, 626951, 626957, 627043, 627047, 627051, 627055, 627351, 627357, 627361, 627365, 627369, 627373, 627381, 627387, 627435, 627439, 627443, 627447, 627453, 627457, 627461, 627465, 627469, 627473, 627477, 627485, 627489, 627495, 627499, 627503, 627507, 627543, 627723, 627727, 627731, 627737, 627741, 627745, 627793, 627801, 627805, 627809, 627815, 627819, 627823, 627827, 627831, 627835, 627839, 627845, 627849, 627853, 627857, 627863, 627867, 627871, 627875, 627879, 627883, 627887, 627891, 627895, 627899, 627905, 627909, 627915, 627975, 627979, 627983, 627987, 627991, 627995, 627999, 628003, 628007, 628011, 628017, 628185, 628401, 628405, 628409, 628413, 628417, 628421, 628923, 628927, 628931, 628935, 628943, 628947, 628951, 628955, 628959, 628963, 628967, 628971, 628977, 628981, 628985, 628989, 629025, 629061, 629065, 629069, 629073, 629077, 629245, 629265, 629269, 629273, 629277, 629313, 629317, 629321, 629325, 629329, 629333, 629337, 629345, 629349, 629355, 629359, 629363, 629367, 629403, 629407, 629419, 629423, 629427, 629433, 629437, 629441, 629445, 629469, 629473, 629477, 629481, 629487, 629511, 629517, 629521, 629525, 629529, 629533, 629725, 629729, 629733, 629745, 629751, 629755, 629759, 629763, 629775, 630715, 630735, 630741, 630745, 630749, 630753, 630757, 630761, 630765, 630769, 630773, 630777, 630781, 630785, 630789, 630793, 630805, 630809, 630813, 630817, 630821, 630843, 630847, 630851, 630855, 630859, 632223, 632563, 632567, 632571, 632575, 632579, 632583, 632587, 632611, 632615, 632619, 632679, 632719, 632723, 632727, 632731, 632735, 632739, 632779, 632783, 632787, 632791, 632795, 632799, 632803, 632807, 632811, 632817, 632821, 632825, 632829, 632833, 632837, 632883, 632887, 632891, 632895, 632901, 632905, 632909, 632915, 632919, 633003, 633007, 633011, 633017, 633021, 633025, 633029, 633033, 633081, 633085, 633089, 633095, 633099, 633103, 633107, 633111, 633115, 633119, 633123, 633127, 633131, 633137, 633141, 633145, 633149, 633155, 633159, 633165, 633169, 633173, 633177, 633181, 633185, 633191, 633195, 633201, 633205, 633309, 633313, 633385, 633389, 633393, 633397, 633409, 633413, 633417, 633421, 633425, 633431, 633435, 633439, 633443, 633447, 633453, 633457, 633465, 633489, 633493, 633561, 633565, 633573, 633577, 633581, 633587, 633591, 633595, 633615, 633619, 633631, 633635, 633639, 633643, 633647, 633669, 633673, 633677, 633681, 633685, 633689, 633693, 633697, 633701, 633705, 633709, 633713, 633717, 633721, 633725, 633729, 633733, 633737, 633743, 633747, 633759, 633763, 633783, 633787, 633795, 633879

Here is a permalink to the algorithm that I used to generate these numbers. Interestingly 26835 is the very first member of the sequence while 633879 is the last in the range up to one million. The graph of the count looks pretty much the same as the one in the Prime Race post. See Figure 1 where is the plot is from 1 to one million with one million marked as 1.0 and 0.6 representing 600,000 etc.

Figure 1: permalink

Yarborough and Anti-Yarborough Primes

I'd not previously heard of a class of primes known as Yarborough primes. My attention was drawn to this class by the fact that 26833, my diurnal age today, is a member of OEIS A296187:

A296187

Yarborough primes that remain Yarborough primes when each of their digits are replaced by their squares.

A Yarborough prime is simply a prime that doesn't contain a zero or a one and clearly 26833 qualifies in that regard. These primes form OEIS A106116:

A106116

Primes with smallest digit > 1.

If we square each of its digits we get 4366499 which is a Yarborough prime. The initial members of the sequence are:

73, 223, 233, 283, 337, 383, 523, 733, 773, 823, 2333, 2683, 2833, 2857, 3323, 3583, 3673, 3733, 3853, 5333, 6673, 6737, 6883, 7333, 7673, 7727, 7877, 8233, 8563, 8623, 22277, 22283, 22727, 23333, 23833, 25237, 25253, 25633, 26227, 26833, 27583, 27827, 27883, 32257

Here is a permalink to a SageMath algorithm that will generate the above sequence. If we can consider squares of digits then why not cubes? This leads to OEIS A296563:

A296563

Yarborough primes that remain Yarborough primes when each of their digits are replaced by their cubes.

The initial members of the sequence are as follows (permalink):

23, 43, 73, 229, 233, 277, 449, 773, 937, 947, 2239, 2243, 2297, 2377, 2777, 3299, 3449, 3727, 3943, 4243, 4423, 4493, 7393, 7723, 7927, 7949, 9227, 9743, 9749, 22277, 22727, 22777, 22943, 23327, 23399, 23497, 23747, 24473, 24733, 27239, 27277, 27427, 27799, 29347, 29443, 29723

There aren't any fourth power Yarborough primes in the range up to one million but there are some fifth power primes in the range up to one million:

683, 2383, 2633, 2663, 6863, 26263, 32833, 36263, 36383, 62233, 63823, 63863, 68633, 68683, 88223, 222883, 232663, 266663, 338383, 386263, 622663, 623683, 632323, 633623, 633883, 663283, 683863, 822223, 828833, 836663, 863833, 866683

The following link mentions the concept of an anti-Yarborough prime and defines it as a prime that contains only zeros and ones e.g. 11 (the first such prime) and 101 (the second such prime). These primes form OEIS A020449:

A020449

Primes whose greatest digit is 1.

The initial members are:

11, 101, 10111, 101111, 1011001, 1100101, 10010101, 10011101, 10100011, 10101101, 10110011, 10111001, 11000111, 11100101, 11110111, 11111101, 100100111, 100111001, 101001001, 101001011, 101100011, 101101111, 101111011, 101111111

I guess the name "Yarborough" derives from bridge where it means "a hand in bridge or whist containing no ace and no card higher than a nine" and is thus useless. The Ace can be assigned the digit 1 and so such a hand would only contain the digits 2 to 9. The name reminds me of a novel that I read in the late sixties called "Yarborough".

Here is an interesting article I found about the book and its author B. H. Friedman.

The 4k-1 versus 4k+1 Prime Race

The number associated with my diurnal age today, 26833, figures in the so-called 4k-1 versus 4k+1 prime race. If we keep a tally of the number of \(4k-1\) primes and \(4k+1\) primes, then eventually certain primes are reached where the two tallies are tied. Here are the list of initial primes where this occurs in the range up to four million:

2, 5, 17, 41, 461, 26833, 26849, 26863, 26881, 26893, 26921, 616769, 616793, 616829, 616843, 616871, 617027, 617257, 617363, 617387, 617411, 617447, 617467, 617473, 617509, 617531, 617579, 617681, 617707, 617719, 618437, 618521, 618593, 618637, 622793, 623171, 623303, 623327, 623351, 623383, 623393, 623431, 623839, 623893, 623929, 623947, 623963, 623983, 624007, 624037, 624049, 624089, 624119, 624139, 624163, 624233, 624251, 626921, 626947, 626959, 627041, 627349, 627383, 627433, 627479, 627491, 627541, 627721, 627811, 627859, 627911, 627973, 628423, 628921, 628939, 629339, 629351, 629381, 629401, 629417, 629483, 629509, 629747, 629773, 630713, 630733, 630823, 630841, 632221, 632561, 632609, 632677, 632717, 632777, 632839, 632881, 632911, 633001, 633091, 633151, 633187, 633427, 633467, 633583, 633613, 633629, 633739, 633757, 633781, 633793, 633799, 633877

As can be seen, 26833 marks the beginning of a cluster of primes (26833, 26849, 26863, 26881, 26893, 26921) after which there is a huge gap until the next cluster (that begins with 616769). Figure 1 shows a graph of the sum of two primes where \(4k-1\) primes are given a value of -1 and \(4k+1\) primes are given a value of +1. When there is an equal number of both types of primes, the sum is zero.

Figure 2: permalink

These primes represent OEIS A007351 and following the link will provides links to other sources of information about this topic.

Look and Count Sequence

On February 10th 2017, I posted about the Look and Say Sequence. After rereading that post, I thought about a variation on that idea and I've called it the Look and Count Sequence. Let's use 1 and example. To begin with there is only one 1 and so we write 11. Now there are two ones and so we write 21. So far it is the same as the Look and Say Sequence.

Here's where it differs. Instead of saying "one two and one one" (1211), we count how many ones, how many twos etc. in order from lowest to highest. Thus 21 becomes "one one and one two" (1112). Now we have "three ones and one two" (3112) which in turn becomes "two ones, one two and one three" (211213). I've written an algorithm to generate the sequence of numbers that result from using 1 as the starting point (permalink). Here it is:

1, 11, 21, 1112, 3112, 211213, 312213, 212223, 114213, 31121314, 41122314, 31221324, 21322314, 21322314

As can be seen the sequence quickly terminates when it reaches 21322314. What about using 2 as the starting point? The result is this sequence enters the one above at the number 1112:

2, 12, 1112, 3112, 211213, 312213, 212223, 114213, 31121314, 41122314, 31221324, 21322314, 21322314

Trying 3, it can be seen that again the sequences overlap and the end result is the same.

3, 13, 1113, 3113, 2123, 112213, 312213, 212223, 114213, 31121314, 41122314, 31221324, 21322314, 21322314

Trying 4, the same end result is reached and even more quickly.

4, 14, 1114, 3114, 211314, 31121314, 41122314, 31221324, 21322314, 21322314 

With 5, the result must be different and it is but the sequence again quickly terminates.

5, 15, 1115, 3115, 211315, 31121315, 41122315, 3122131415, 4122231415, 3132132415, 3122331415, 3122331415

When we investigate 6, 7, 8 and 9, it can be seen that the pattern in the same.

6, 16, 1116, 3116, 211316, 31121316, 41122316, 3122131416, 4122231416, 3132132416, 3122331416, 3122331416

7, 17, 1117, 3117, 211317, 31121317, 41122317, 3122131417, 4122231417, 3132132417, 3122331417, 3122331417

8, 18, 1118, 3118, 211318, 31121318, 41122318, 3122131418, 4122231418, 3132132418, 3122331418, 3122331418

9, 19, 1119, 3119, 211319, 31121319, 41122319, 3122131419, 4122231419, 3132132419, 3122331419, 3122331419

What if we take an arbitrary and larger number, let's say 78651154? The result is a two step loop (5142131415261718 --> 6122132425161718 --> 5142131415261718):

78651154, 211425161718, 51221415161718, 61221425161718, 51321415261718, 5122131425161718, 6132131425161718, 6122231415261718, 5142131415261718, 6122132425161718, 5142131415261718

Even if we take an unusual number like 999999999999, the result is also a two step loop:

999999999999, 129, 111219, 411219, 31121419, 4112131419, 5112132419, 412213141519, 512213241519, 413213142519, 412223241519, 314213241519, 412223241519

So far I've not considered numbers with 0 as a digital. If we try 1004056906, the result is the same two step loop:

1004056906, 401114152619, 10511224151619, 10612214251619, 10513214152619, 1051221314251619, 1061321314251619, 1061222314152619, 1051421314152619, 1061221324251619, 1051421314152619

So it seems that, no matter what the starting number, the sequence of terms generated quickly terminates. This is true for even large numbers like:

101111011101100222222222222222222222222222222222

The steps leading to another two step loop are:

1. 101111011101100222222222222222222222222222222222
2. 50101332
3. 2021122315
4. 1031421315
5. 104112231415
6. 105122132415
7. 104132131425
8. 104122232415
9. 103142132415
10. 104122232415

It's interesting to explore what numbers set records for the number of steps required before they terminate. Here is what I found for the range of numbers up to 100,000 (permalink):

1 requires 13 steps

60 requires 14 steps

70 requires 15 steps

80 requires 16 steps

109 requires 17 steps

2008 requires 18 steps

2009 requires 20 steps

9009 requires 21 steps

What's Special About 3435?

The number 3435 has the honour of being the only base 10 Munchausen number. So what is a Munchausen number? Wikipedia provides this definition:

A Munchausen number is a natural number in a given number base \(b\) that is equal to the sum of its digits each raised to the power of itself. An example in base 10 is 3435, because \(3435=3^3+4^4+3^3+5^5\). The term "Munchausen number" was coined by Dutch mathematician and software engineer Daan van Berkel in 2009, as this evokes the story of Baron Munchausen raising himself up by his own ponytail because each digit is raised to the power of itself.

Of course, the number 1 qualifies as well but this is trivial and can be ignored. The only rival to 3435 comes in the form of 438579088 but runs into the problem of what the value of \(0^0\) is. If we take \(0^0=0\) then it does qualify because:$$438579088 = 4^4+3^3+8^8+5^5+7^7+9^9+0^0+8^8+8^8$$However, \(0^0\) is also commonly evaluated as 1 (see link) and so there is an ambiguity surrounding 438579088. If we take \(0^0=0\) then it does qualify as a Munchausen number but if we take \(0^0=1\), it doesn't. 3425 however, suffers from no such ambiguity.

Let's spell out 3435's unique property in large type:

\(3435=3^3+4^4+3^3+5^5\)

There are Munchausen numbers in other number bases as well but in this post I'm just focusing on base 10. Here is a permalink for identifying Munchausen numbers in base 10.

A Special Class of Semiprimes

Looking at the factors of my diurnal age today, I was immediately struck by the fact that each factor was a permutation of the digits of the other. To be specific:

26827 = 139 * 193

I immediately speculated as to how common such an occurrence was. To begin my investigation, I had to exclude square semiprimes such as 121 because they qualify trivially. Here is a permalink to the algorithm that I developed to find all such semiprimes up to one million. Below are the semiprimes along with their factorisation:

403 = 13 * 31

1207 = 17 * 71

2701 = 37 * 73

7663 = 79 * 97

14803 = 113 * 131

23701 = 137 * 173

26827 = 139 * 193

34417 = 127 * 271

35143 = 113 * 311

35263 = 179 * 197

40741 = 131 * 311

43429 = 137 * 317

54841 = 173 * 317

62431 = 149 * 419

70027 = 239 * 293

73159 = 149 * 491

75007 = 107 * 701

89647 = 157 * 571

99919 = 163 * 613

101461 = 241 * 421

102853 = 163 * 631

103039 = 167 * 617

103603 = 313 * 331

117907 = 157 * 751

125701 = 337 * 373

127087 = 167 * 761

128701 = 179 * 719

130771 = 251 * 521

140209 = 149 * 941

141643 = 197 * 719

146791 = 181 * 811

150463 = 379 * 397

153211 = 349 * 439

173809 = 179 * 971

174001 = 191 * 911

182881 = 199 * 919

191287 = 197 * 971

197209 = 199 * 991

201379 = 277 * 727

205729 = 419 * 491

212887 = 359 * 593

230701 = 281 * 821

232909 = 283 * 823

246991 = 367 * 673

247021 = 337 * 733

249979 = 457 * 547

257821 = 347 * 743

273409 = 373 * 733

280081 = 379 * 739

293383 = 397 * 739

295501 = 461 * 641

297709 = 463 * 643

302149 = 467 * 647

326371 = 389 * 839

342127 = 359 * 953

355123 = 379 * 937

367639 = 563 * 653

371989 = 397 * 937

374971 = 569 * 659

382387 = 389 * 983

386803 = 613 * 631

394279 = 419 * 941

427729 = 619 * 691

428821 = 571 * 751

436789 = 577 * 757

453613 = 479 * 947

462031 = 491 * 941

469537 = 617 * 761

503059 = 587 * 857

565129 = 593 * 953

589429 = 683 * 863

643063 = 709 * 907

690199 = 787 * 877

692443 = 739 * 937

698149 = 719 * 971

743623 = 769 * 967

778669 = 797 * 977

824737 = 839 * 983

910729 = 919 * 991

There are 79 semiprimes in total. Here is the list (I'm surprised this sequence hasn't made it into the OEIS but it hasn't and I've no intention of submitting it):

403, 1207, 2701, 7663, 14803, 23701, 26827, 34417, 35143, 35263, 40741, 43429, 54841, 62431, 70027, 73159, 75007, 89647, 99919, 101461, 102853, 103039, 103603, 117907, 125701, 127087, 128701, 130771, 140209, 141643, 146791, 150463, 153211, 173809, 174001, 182881, 191287, 197209, 201379, 205729, 212887, 230701, 232909, 246991, 247021, 249979, 257821, 273409, 280081, 293383, 295501, 297709, 302149, 326371, 342127, 355123, 367639, 371989, 374971, 382387, 386803, 394279, 427729, 428821, 436789, 453613, 462031, 469537, 503059, 565129, 589429, 643063, 690199, 692443, 698149, 743623, 778669, 824737, 910729

As can be seen, 26827 is only the seventh such number. I may well be dead before I see the next such semiprime (34417) that will represent my diurnal age on June 26th 2043. If I'm still here I would have passed my 94th birthday.

Another approach to finding these semiprimes would be to test all the primes in a given range, find what permutations of the digits produce primes and multiply the two together. However, this doesn't make for a very efficient algorithm. When I tried it on SageMathCell, the program timed out so the algorithm linked to in the permalink is far more efficient, producing a speedy output.

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.