Five Five Numbers

The number associated with my diurnal age today, \( \textbf{27905} \), is one of those numbers that it is difficult to find anything of interest about. However, as usual, a little investigation turned up something special about it. It is what I call a five five number meaning it meets the following criteria:

• it is a composite and squarefree number
• its digits contain a single 5
• each prime factor contains at least one 5 for a total of three 5's
• its arithmetical digital root is 5
  
  

In the range up to 40000, there are only six numbers that satisfy these criteria and they are 12785, 27635, 27815, 27905, 28265 and 32765. Here are the details (permalink):

 number   factors    root

  12785    5 * 2557   5

  27635    5 * 5527   5

  27815    5 * 5563   5

  27905    5 * 5581   5

  28265    5 * 5653   5

  32765    5 * 6553   5

You could extend this idea to digits other than \( \textbf{5} \). The digit \( \textbf{3} \) produces too many suitable numbers but what about the digit \( \textbf{4} \) where we require this of the number:

• it is a composite and squarefree number
• its digits contain a single 4
• each prime factor contains at least one 4 for a total of two 4's
• its arithmetical digital root is 4
  
  

In the range up to 40000, only one number satisfies and that is 19147 = 41 x 467 with a digital root of 4 (permalink). Nothing for the digit \( \textbf{6} \) up to one million. For the digit \( \textbf{7} \) there is only one number up to one million and that is 544579 = 7 x 77797 with a digital root of 7 - we require that its prime factors contain a total of five 7's (permalink). For the digits \( \textbf{8} \) and \( \textbf{9} \), no numbers qualify up to one million.

So it turns out that 27905 is not so uninteresting after all. Additionally, it is a \( \textbf{Proth} \) number, since it is equal to \(109 \times 2^8 + 1\) and \(109 < 28 \). I've written about these before in a post titled Proth Numbers.

Some Special Sphenic Numbers

Today I turned \( \textbf{27782} \) days old and one of the properties of 27782 is that it is sphenic since:$$27782=2 \times 29 \times 479$$However, looking at the digital roots of the number and its factor we notice an interesting fact:$$ \underbrace{27782}_{8}=\underbrace{2}_{2} \times \underbrace{29}_{2} \times \underbrace{479}_{2}$$The respective digital roots are shown under the number and its factors and we see that the digital roots of the factors (2) multiply together to give the digital root of the number (8). The only other way this can occur is if all the digital roots are 1.

In the range up to 100,000 there are 230 sphenic numbers with the property that the digital roots of the factors multiply to give the digital root of the number. These numbers are (permalink):

638, 1034, 1826, 2222, 2726, 3014, 3806, 4202, 4814, 4994, 5786, 5858, 6182, 6974, 7766, 7802, 7946, 8558, 9494, 9746, 10034, 10142, 10538, 11078, 12518, 12878, 12914, 13166, 14102, 14498, 14894, 14993, 15254, 16262, 16298, 16766, 17954, 18062, 18386, 18458, 18854, 20042, 20438, 20474, 20834, 21338, 21626, 22418, 22562, 22742, 24002, 24398, 24722, 25586, 25694, 25982, 26414, 26477, 26738, 26774, 27674, 27782, 28358, 28718, 28754, 29798, 29942, 31526, 31706, 31922, 32219, 32714, 33002, 33182, 33506, 34046, 34298, 34946, 35486, 36566, 36674, 37178, 37682, 37862, 38222, 38582, 39266, 39842, 40634, 41642, 41822, 42911, 43334, 43406, 43658, 43703, 44594, 45026, 45386, 45782, 45854, 46178, 46646, 47366, 47402, 47618, 48554, 48662, 49346, 49706, 50534, 51319, 51326, 51722, 52217, 52334, 52622, 52838, 53126, 53306, 53486, 53702, 53882, 54098, 54494, 54926, 55178, 55187, 55682, 56078, 56762, 57014, 57662, 58454, 58598, 59102, 59246, 59642, 60038, 60254, 60929, 61622, 61946, 62018, 62198, 62414, 63278, 63638, 63998, 64034, 64322, 64394, 64574, 65186, 65978, 66086, 67454, 67958, 68498, 70586, 70829, 71306, 71522, 72062, 72413, 73106, 73538, 73898, 74762, 75086, 75806, 75878, 76274, 76526, 76627, 76994, 77174, 77858, 78254, 78542, 78578, 78866, 78938, 79046, 79514, 80558, 81422, 81818, 83114, 83897, 84158, 84986, 85382, 85634, 86174, 86246, 86714, 86858, 87326, 87758, 88154, 88334, 89018, 89281, 89441, 89486, 89639, 89738, 90422, 90926, 90998, 92213, 92402, 93122, 93302, 93554, 93698, 94454, 95678, 95786, 96686, 96722, 96758, 97226, 97262, 97442, 98054, 98747, 98846, 99818

However, of these only three have digital roots that are all equal to 1. These are:$$ \begin{align} 51319 &= 19 \times 37 \times 73\\76627 &= 19 \times 37 \times 109\\ 89281 &= 19 \times 37 \times 127 \end{align} $$As can be seen, two of the factors (19 and 37) are the same for all three numbers. If we extend the range to one million, these two factor and 73 (the reversal of 37) make frequent appearances.

If we relax the requirement that the digital roots of the factors must be equal and require only the the digital roots of the factors multiply together to give the digital root of the number, then we find that 1969 numbers satisfy and that will include the 230 numbers mentioned earlier (permalink). An example would be 27813:$$ \underbrace{27813}_{3} = \underbrace{3}_{3} \times \underbrace{73}_{1} \times \underbrace{127}_{1}$$

Consolidating Fibonacci-like Numbers

In my post titled Additive Fibonacci-like Numbers I was dealing with additive digital roots to generate additional digits after the starting two digits were in place. For example, let's start with 78:$$ \begin{align} 78 \rightarrow 7 + 8 =15 \rightarrow 1+5=6 &\rightarrow 786 \\786 \rightarrow 8+6=14 \rightarrow 1+4=5 &\rightarrow 7865 \\7865 \rightarrow 6+5 =11 \rightarrow 1+1=2 &\rightarrow 78652 \end{align} $$We could keep going forever. The advantage of this approach is that the sum of the two previous digits reduces to a single digit between 1 and 9. Let's call these types of numbers Additive Fibonacci-like Numbers of the First Type. Between 100 and 1,000,000 these numbers are:

101, 112, 123, 134, 145, 156, 167, 178, 189, 191, 202, 213, 224, 235, 246, 257, 268, 279, 281, 292, 303, 314, 325, 336, 347, 358, 369, 371, 382, 393, 404, 415, 426, 437, 448, 459, 461, 472, 483, 494, 505, 516, 527, 538, 549, 551, 562, 573, 584, 595, 606, 617, 628, 639, 641, 652, 663, 674, 685, 696, 707, 718, 729, 731, 742, 753, 764, 775, 786, 797, 808, 819, 821, 832, 843, 854, 865, 876, 887, 898, 909, 911, 922, 933, 944, 955, 966, 977, 988, 999, 1011, 1123, 1235, 1347, 1459, 1562, 1674, 1786, 1898, 1911, 2022, 2134, 2246, 2358, 2461, 2573, 2685, 2797, 2819, 2922, 3033, 3145, 3257, 3369, 3472, 3584, 3696, 3718, 3821, 3933, 4044, 4156, 4268, 4371, 4483, 4595, 4617, 4729, 4832, 4944, 5055, 5167, 5279, 5382, 5494, 5516, 5628, 5731, 5843, 5955, 6066, 6178, 6281, 6393, 6415, 6527, 6639, 6742, 6854, 6966, 7077, 7189, 7292, 7314, 7426, 7538, 7641, 7753, 7865, 7977, 8088, 8191, 8213, 8325, 8437, 8549, 8652, 8764, 8876, 8988, 9099, 9112, 9224, 9336, 9448, 9551, 9663, 9775, 9887, 9999, 10112, 11235, 12358, 13472, 14595, 15628, 16742, 17865, 18988, 19112, 20224, 21347, 22461, 23584, 24617, 25731, 26854, 27977, 28191, 29224, 30336, 31459, 32573, 33696, 34729, 35843, 36966, 37189, 38213, 39336, 40448, 41562, 42685, 43718, 44832, 45955, 46178, 47292, 48325, 49448, 50551, 51674, 52797, 53821, 54944, 55167, 56281, 57314, 58437, 59551, 60663, 61786, 62819, 63933, 64156, 65279, 66393, 67426, 68549, 69663, 70775, 71898, 72922, 73145, 74268, 75382, 76415, 77538, 78652, 79775, 80887, 81911, 82134, 83257, 84371, 85494, 86527, 87641, 88764, 89887, 90999, 91123, 92246, 93369, 94483, 95516, 96639, 97753, 98876, 99999

However, in my previous post, Variations on the Taxi Cab Number, I was not working with the digital roots and this is a severe limitation. The early digits need to be small if the digits are to progress in a Fibonacci-like manner. That's why, in the range of numbers, up to one million, the largest number is 303369. This number is constructed as follows beginning with the first two digits 3 and 0:$$ \begin{align} 30 \rightarrow 3 + 0 &= 3 \rightarrow 303 \\ 303 \rightarrow 0+3 &=3 \rightarrow 3033\\3033 \rightarrow 3 + 3 &= 6 \rightarrow 30336\\30336 \rightarrow 3+6 &= 9 \rightarrow 303369 \end{align}$$We can't go any further because of the final two digits: 6 + 9 = 15. Let's call these types of numbers Additive Fibonacci-like Numbers of the Second Type. Between 100 and 1,000,000 these numbers are:

101, 112, 123, 134, 145, 156, 167, 178, 189, 202, 213, 224, 235, 246, 257, 268, 279, 303, 314, 325, 336, 347, 358, 369, 404, 415, 426, 437, 448, 459, 505, 516, 527, 538, 549, 606, 617, 628, 639, 707, 718, 729, 808, 819, 909, 1011, 1123, 1235, 1347, 1459, 2022, 2134, 2246, 2358, 3033, 3145, 3257, 3369, 4044, 4156, 4268, 5055, 5167, 5279, 6066, 6178, 7077, 7189, 8088, 9099, 10112, 11235, 12358, 20224, 21347, 30336, 31459, 40448, 101123, 112358, 202246, 303369

With bases higher than 10, the 1 to 9 digit limitation can be exceeded. For example in base 16, if we start as before with an initial 78 then a third digit is possible:$$78 \rightarrow 7 + 8 = 15 = F \rightarrow 78F$$Thus we have:$$ \begin{align} 78F_{16} &= 7 \times 16^2 + 8 \times 16 + 15 \\ &=1935_{10} \end{align} $$This means that 1935 is an Additive Fibonacci-like Number of the Second Type in base 16. Here is a list of numbers greater than 27000 and less than 40000 that are "additive Fibonacci-like" and of the "second type" in base 16 (permalink):

• 28791 --> 7077
• 29065 --> 7189
• 29339 --> 729b
• 29613 --> 73ad
• 29887 --> 74bf
• 32904 --> 8088
• 33178 --> 819a
• 33452 --> 82ac
• 33726 --> 83be
• 37017 --> 9099
• 37291 --> 91ab
• 37565 --> 92bd
• 37839 --> 93cf

Additive Fibonacci-like Numbers of the Second Type in base 16 are thus:

28791, 29065, 29339, 29613, 29887, 32904, 33178, 33452, 33726, 37017, 37291, 37565, 37839

*****************************

Here the numbers greater than 27000 and less than 40000 for base 15:

• 27128 --> 8088
• 27370 --> 819a
• 27612 --> 82ac
• 27854 --> 83be
• 30519 --> 9099
• 30761 --> 91ab
• 31003 --> 92bd
• 33910 --> a0aa
• 34152 --> a1bc
• 34394 --> a2ce
• 37301 --> b0bb
• 37543 --> b1cd

Additive Fibonacci-like Numbers of the Second Type in base 15 are (permalink):

27128, 27370, 27612, 27854, 30519, 30761, 31003, 33910, 34152, 34394, 37301, 37543

*****************************

Here are the numbers greater than 27000 and less than 40000 for base 14 (permalink): 

• 27590 --> a0aa
• 27802 --> a1bc
• 30349 --> b0bb
• 30561 --> b1cd
• 33108 --> c0cc
• 35867 --> d0dd
• 38628 --> 10112

Additive Fibonacci-like Numbers of the Second Type in base 14 are thus:

27590, 27802, 30349, 30561, 33108, 35867, 38628

*****************************

Here are the numbers greater than 27000 and less than 40000 for base 13 (permalink):

• 28745 --> 10112
• 31140 --> 11235
• 33535 --> 12358
• 35930 --> 1347b

Additive Fibonacci-like Numbers of the Second Type in base 13 are thus:

28745, 31140, 33535, 35930

*****************************

For base 12, there are none between 27000 and 40000 but for base 11 we have (permalink):

• 29550 --> 20224
• 31027 --> 21347
• 32504 --> 2246a

Additive Fibonacci-like Numbers of the Second Type in base 11 are thus:

29550, 31027, 32504

*****************************

Just for completeness I'll now look at bases 10 and lower. For base 10, we have (permalink):

• 30336 --> 30336
• 31459 --> 31459

Additive Fibonacci-like Numbers of the Second Type in base 10 are thus:

30336, 31459

*****************************

For base 9 there are none but for base 8 there is one (permalink):

• 33363 --> 101123

Additive Fibonacci-like Numbers of the Second Type in base 8 are thus :

33363

*****************************

For base 7, we have 

• 34432 --> 202246

Additive Fibonacci-like Numbers of the Second Type in base 7 are thus:

34432

There are no suitable numbers in the range 27000 to 40000 for bases 2, 3, 4, 5 and 6. I've added this determination of whether a number is additive Fibonacci-like of the second type to my multipurpose algorithm.

RIGHT TO LEFT INSTEAD OF LEFT TO RIGHT

There's no compulsion to proceed from left to right when working with digits and so a new set of numbers can be generated by simply reversing the order of the digits. Thus Additive Fibonacci-like Numbers of the First Type are shown below where digit progression is from right to left:

101, 119, 128, 137, 146, 155, 164, 173, 182, 191, 202, 211, 229, 238, 247, 256, 265, 274, 283, 292, 303, 312, 321, 339, 348, 357, 366, 375, 384, 393, 404, 413, 422, 431, 449, 458, 467, 476, 485, 494, 505, 514, 523, 532, 541, 559, 568, 577, 586, 595, 606, 615, 624, 633, 642, 651, 669, 678, 687, 696, 707, 716, 725, 734, 743, 752, 761, 779, 788, 797, 808, 817, 826, 835, 844, 853, 862, 871, 889, 898, 909, 918, 927, 936, 945, 954, 963, 972, 981, 999, 1101, 1191, 1283, 1375, 1467, 1559, 1642, 1734, 1826, 1918, 2119, 2202, 2292, 2384, 2476, 2568, 2651, 2743, 2835, 2927, 3128, 3211, 3303, 3393, 3485, 3577, 3669, 3752, 3844, 3936, 4137, 4229, 4312, 4404, 4494, 4586, 4678, 4761, 4853, 4945, 5146, 5238, 5321, 5413, 5505, 5595, 5687, 5779, 5862, 5954, 6155, 6247, 6339, 6422, 6514, 6606, 6696, 6788, 6871, 6963, 7164, 7256, 7348, 7431, 7523, 7615, 7707, 7797, 7889, 7972, 8173, 8265, 8357, 8449, 8532, 8624, 8716, 8808, 8898, 8981, 9182, 9274, 9366, 9458, 9541, 9633, 9725, 9817, 9909, 9999, 11918, 12835, 13752, 14678, 15505, 15595, 16422, 17348, 18265, 19182, 21101, 21191, 22927, 23844, 24761, 25687, 26514, 27431, 28357, 29274, 31283, 32119, 33936, 34853, 35779, 36606, 36696, 37523, 38449, 39366, 41375, 42202, 42292, 43128, 44945, 45862, 46788, 47615, 48532, 49458, 51467, 52384, 53211, 54137, 55954, 56871, 57707, 57797, 58624, 59541, 61559, 62476, 63303, 63393, 64229, 65146, 66963, 67889, 68716, 69633, 71642, 72568, 73485, 74312, 75238, 76155, 77972, 78808, 78898, 79725, 81734, 82651, 83577, 84404, 84494, 85321, 86247, 87164, 88981, 89817, 91826, 92743, 93669, 94586, 95413, 96339, 97256, 98173, 99909, 99999

Similarly Additive Fibonacci-like Numbers of the Second Type are shown below where digit progression is from right to left:

101, 202, 211, 303, 312, 321, 404, 413, 422, 431, 505, 514, 523, 532, 541, 606, 615, 624, 633, 642, 651, 707, 716, 725, 734, 743, 752, 761, 808, 817, 826, 835, 844, 853, 862, 871, 909, 918, 927, 936, 945, 954, 963, 972, 981, 1101, 2202, 3211, 3303, 4312, 4404, 5321, 5413, 5505, 6422, 6514, 6606, 7431, 7523, 7615, 7707, 8532, 8624, 8716, 8808, 9541, 9633, 9725, 9817, 9909, 21101, 42202, 53211, 63303, 74312, 84404, 85321, 95413, 321101, 642202, 853211, 963303

If we proceed from left to right, then the third digit is the difference between the first and second digits and so on (this is the subtraction sequence mentioned in my previous post).

Palindromic Day 27572

Every 100 days another palindrome day rolls by and yesterday I celebrated palindromic day 27572. Now this palindrome has an arithmetic digital root that is equal to its central digit of 5. This is because:$$ 27572 \rightarrow 2+7+5+7+2= 23 \rightarrow 2 + 3 =5 $$However, the absolute difference between the first two digits (and of course the last two digits as well) is also equal to the digital root and the central digit.

| 2 - 7 | = 5 = | 7 - 2 |

This makes the palindrome extra special and in the range of five digit numbers from 10000 to 99999 only the following palindromes have the properties previously mentioned. These are (permalink):

18781, 27572, 36363, 45154, 54145, 63336, 72527, 81718, 90909

If we allowed leading zeros then we would have:

• 09990 has the same digits as 90909

| 0 - 9 | = 9 = | 9 - 0 |

• 18781 has the same digits as 81718

| 1 - 8 | = 7 = | 8 - 1 |

• 27572 has the same digits as 72527

| 2 - 7 | = 5 = | 7 - 2 |

• 36363 has the same digits as 63336

| 3 - 6 | = 3 = | 6 - 3 |

• 45154 has the same digits as 54145

| 4 - 5 | = 1 = | 5 - 4 |

Remember that the central digit is also the arithmetic digital root of the number.

New Telephone Number

It's always exciting to get a new telephone number because of the properties of the number that may turn up. Having arrived in Australia for a temporary stay, I needed a telephone number and the number that I was given was:$$0451591949 \rightarrow 451591949$$Now this number is prime but has additional "primeness" embedded in it because:$$ \text{Sum of digits is }47 \text{ and prime}\\47 \rightarrow 4 + 7 = 11 \\ 11 \rightarrow 1+1=2 \\ \text{ 2 (the digital root) is prime}$$Now 1949 and 1951 form a pair of twin primes but it turns out that:$$ 451591949 \text{ and } 4511591951 \\ \text{ also form a pair of twin primes}$$Furthermore:$$451591949 \text{ is a Sophie Germain prime} \\ 451591949 \times 2 + 1 = 903183899 \text{ a prime}$$\(451591949\) is also a Chen prime defined as a prime number  \(p\)  such that  \(p+2\)  is either a prime or a semiprime. Jing Run Chen, after which they are named, proved in 1966 that there are infinitely many such primes. Binbin Zhou has proved in 2009 that the Chen primes contain arbitrarily long arithmetic progressions.

So overall I'm very happy with my new prime telephone number even though it will lapse and be discarded once I leave the country. For now though it's mine and prime!

Additive Fibonacci-like Numbers

Consider all two digit numbers from 10 to 99 and use these as the seed digits that will generate a third digit by ADDITION of the two digits and by then finding the DIGITAL ROOT of the resultant sum. Here are the 90 starting numbers.

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

These 90 two digit numbers will generate another 90 three digit numbers. These are:

101, 112, 123, 134, 145, 156, 167, 178, 189, 191, 202, 213, 224, 235, 246, 257, 268, 279, 281, 292, 303, 314, 325, 336, 347, 358, 369, 371, 382, 393, 404, 415, 426, 437, 448, 459, 461, 472, 483, 494, 505, 516, 527, 538, 549, 551, 562, 573, 584, 595, 606, 617, 628, 639, 641, 652, 663, 674, 685, 696, 707, 718, 729, 731, 742, 753, 764, 775, 786, 797, 808, 819, 821, 832, 843, 854, 865, 876, 887, 898, 909, 911, 922, 933, 944, 955, 966, 977, 988, 999

These in turn will produce 90 four digit numbers. These are:

1011, 1123, 1235, 1347, 1459, 1562, 1674, 1786, 1898, 1911, 2022, 2134, 2246, 2358, 2461, 2573, 2685, 2797, 2819, 2922, 3033, 3145, 3257, 3369, 3472, 3584, 3696, 3718, 3821, 3933, 4044, 4156, 4268, 4371, 4483, 4595, 4617, 4729, 4832, 4944, 5055, 5167, 5279, 5382, 5494, 5516, 5628, 5731, 5843, 5955, 6066, 6178, 6281, 6393, 6415, 6527, 6639, 6742, 6854, 6966, 7077, 7189, 7292, 7314, 7426, 7538, 7641, 7753, 7865, 7977, 8088, 8191, 8213, 8325, 8437, 8549, 8652, 8764, 8876, 8988, 9099, 9112, 9224, 9336, 9448, 9551, 9663, 9775, 9887, 9999

These in turn will produce 90 five digit numbers. These are:

10112, 11235, 12358, 13472, 14595, 15628, 16742, 17865, 18988, 19112, 20224, 21347, 22461, 23584, 24617, 25731, 26854, 27977, 28191, 29224, 30336, 31459, 32573, 33696, 34729, 35843, 36966, 37189, 38213, 39336, 40448, 41562, 42685, 43718, 44832, 45955, 46178, 47292, 48325, 49448, 50551, 51674, 52797, 53821, 54944, 55167, 56281, 57314, 58437, 59551, 60663, 61786, 62819, 63933, 64156, 65279, 66393, 67426, 68549, 69663, 70775, 71898, 72922, 73145, 74268, 75382, 76415, 77538, 78652, 79775, 80887, 81911, 82134, 83257, 84371, 85494, 86527, 87641, 88764, 89887, 90999, 91123, 92246, 93369, 94483, 95516, 96639, 97753, 98876, 99999

Forgetting about the original two digit numbers, let's group all the three, four and five digits number together so that we have 270 numbers. These are:

101, 112, 123, 134, 145, 156, 167, 178, 189, 191, 202, 213, 224, 235, 246, 257, 268, 279, 281, 292, 303, 314, 325, 336, 347, 358, 369, 371, 382, 393, 404, 415, 426, 437, 448, 459, 461, 472, 483, 494, 505, 516, 527, 538, 549, 551, 562, 573, 584, 595, 606, 617, 628, 639, 641, 652, 663, 674, 685, 696, 707, 718, 729, 731, 742, 753, 764, 775, 786, 797, 808, 819, 821, 832, 843, 854, 865, 876, 887, 898, 909, 911, 922, 933, 944, 955, 966, 977, 988, 999, 1011, 1123, 1235, 1347, 1459, 1562, 1674, 1786, 1898, 1911, 2022, 2134, 2246, 2358, 2461, 2573, 2685, 2797, 2819, 2922, 3033, 3145, 3257, 3369, 3472, 3584, 3696, 3718, 3821, 3933, 4044, 4156, 4268, 4371, 4483, 4595, 4617, 4729, 4832, 4944, 5055, 5167, 5279, 5382, 5494, 5516, 5628, 5731, 5843, 5955, 6066, 6178, 6281, 6393, 6415, 6527, 6639, 6742, 6854, 6966, 7077, 7189, 7292, 7314, 7426, 7538, 7641, 7753, 7865, 7977, 8088, 8191, 8213, 8325, 8437, 8549, 8652, 8764, 8876, 8988, 9099, 9112, 9224, 9336, 9448, 9551, 9663, 9775, 9887, 9999, 10112, 11235, 12358, 13472, 14595, 15628, 16742, 17865, 18988, 19112, 20224, 21347, 22461, 23584, 24617, 25731, 26854, 27977, 28191, 29224, 30336, 31459, 32573, 33696, 34729, 35843, 36966, 37189, 38213, 39336, 40448, 41562, 42685, 43718, 44832, 45955, 46178, 47292, 48325, 49448, 50551, 51674, 52797, 53821, 54944, 55167, 56281, 57314, 58437, 59551, 60663, 61786, 62819, 63933, 64156, 65279, 66393, 67426, 68549, 69663, 70775, 71898, 72922, 73145, 74268, 75382, 76415, 77538, 78652, 79775, 80887, 81911, 82134, 83257, 84371, 85494, 86527, 87641, 88764, 89887, 90999, 91123, 92246, 93369, 94483, 95516, 96639, 97753, 98876, 99999

Viewed as a Fibonacci-like sequence, the sequence of digits will eventually cycle. Take 27977 as an example. The progression is:$$2, 7, 9, 7, 7, 5, 3, 8, 2, 1, 3, 4, 7, 2, 9, 2, 2, 4, 6, 1, 7, 8, 6, 5, 2, 7, 9, 7, 7, \dots $$An alternative to this progression of digits is to determine the arithmetical digital root of the cumulative sum of digits and use this as the next digit. Here is a permalink that will generate this sequence of 270 numbers. Here are the numbers:

101, 112, 123, 134, 145, 156, 167, 178, 189, 191, 202, 213, 224, 235, 246, 257, 268, 279, 281, 292, 303, 314, 325, 336, 347, 358, 369, 371, 382, 393, 404, 415, 426, 437, 448, 459, 461, 472, 483, 494, 505, 516, 527, 538, 549, 551, 562, 573, 584, 595, 606, 617, 628, 639, 641, 652, 663, 674, 685, 696, 707, 718, 729, 731, 742, 753, 764, 775, 786, 797, 808, 819, 821, 832, 843, 854, 865, 876, 887, 898, 909, 911, 922, 933, 944, 955, 966, 977, 988, 999, 1012, 1124, 1236, 1348, 1451, 1563, 1675, 1787, 1899, 1912, 2024, 2136, 2248, 2351, 2463, 2575, 2687, 2799, 2812, 2924, 3036, 3148, 3251, 3363, 3475, 3587, 3699, 3712, 3824, 3936, 4048, 4151, 4263, 4375, 4487, 4599, 4612, 4724, 4836, 4948, 5051, 5163, 5275, 5387, 5499, 5512, 5624, 5736, 5848, 5951, 6063, 6175, 6287, 6399, 6412, 6524, 6636, 6748, 6851, 6963, 7075, 7187, 7299, 7312, 7424, 7536, 7648, 7751, 7863, 7975, 8087, 8199, 8212, 8324, 8436, 8548, 8651, 8763, 8875, 8987, 9099, 9112, 9224, 9336, 9448, 9551, 9663, 9775, 9887, 9999, 10124, 11248, 12363, 13487, 14512, 15636, 16751, 17875, 18999, 19124, 20248, 21363, 22487, 23512, 24636, 25751, 26875, 27999, 28124, 29248, 30363, 31487, 32512, 33636, 34751, 35875, 36999, 37124, 38248, 39363, 40487, 41512, 42636, 43751, 44875, 45999, 46124, 47248, 48363, 49487, 50512, 51636, 52751, 53875, 54999, 55124, 56248, 57363, 58487, 59512, 60636, 61751, 62875, 63999, 64124, 65248, 66363, 67487, 68512, 69636, 70751, 71875, 72999, 73124, 74248, 75363, 76487, 77512, 78636, 79751, 80875, 81999, 82124, 83248, 84363, 85487, 86512, 87636, 88751, 89875, 90999, 91124, 92248, 93363, 94487, 95512, 96636, 97751, 98875, 99999

Let's take 26875 as an example. We begin with 26 as our seed number and then proceed thus: $$ \begin{align} 26 \text{ has digit sum } 8 &\rightarrow 268 \\ 268 \text{ has digit sum } 16 \equiv 7 &\rightarrow 2687 \\ 2687 \text{ has digit sum } 23 \equiv 5 &\rightarrow 26875 \end{align} $$The three digit numbers are the same as earlier but the differences arise in the four and five digit numbers. Let's compare the previous cumulative results with the seed number 26 again but using the earlier two digit approach:$$ \begin{align} 26 \text{ has digit sum } 8 &\rightarrow 268 \\ 68 \text{ has digit sum } 14 \equiv 5 &\rightarrow 2685 \\ 85 \text{ has digit sum } 13 \equiv 4 &\rightarrow 26854 \end{align} $$

A Multiplicity of Digits: Part 2

A variation on the theme of my previous post, that also involves the multiple occurrence of the same digits, are these numbers that comprise a sequence that I've referenced as S107 in my Bespoken for Sequences database. 

Sphenic numbers containing the digit 3 whose three prime factors also contain the digit 3 and whose additive digital root is 3.

The first example of such a number is 1443 = 3 * 13 * 37 with a digital root of 3. There are 61 such numbers in the range up to 40000. Here is the list (permalink):

1443, 3441, 3657, 3999, 4773, 6357, 8103, 9039, 9453, 11037, 11433, 11937, 11973, 13143, 13197, 13287, 13611, 14313, 15483, 17931, 18093, 20397, 20739, 21423, 21783, 21873, 23907, 23943, 24357, 24753, 26319, 28137, 29739, 30441, 30567, 30783, 31341, 31413, 32097, 32457, 32619, 33267, 34077, 34113, 34131, 34437, 34689, 34707, 34743, 35247, 35697, 36507, 36543, 36741, 36921, 37047, 37407, 38001, 38739, 38847, 39603

A twist on this theme is to consider sphenic numbers that do NOT contain the digit 3. Such numbers could be considered as having a hidden multiplicity of digits because the prevalence of the digit is not immediately obvious. The same could be said of the numbers just mentioned but those cases the repeating digit is overtly visible. Here is the revised criteria:

Sphenic numbers NOT containing the digit 3 whose three prime factors also contain the digit 3 and whose additive digital root is 3.

The first such number is 1209 = 3 * 13 * 31 with a digital root of 3. There are 45 such numbers in the range up to 40000. Here they are (permalink):

1209, 1677, 2847, 4017, 5421, 5727, 6789, 7527, 7797, 8697, 9417, 9579, 12207, 12909, 12927, 14547, 14781, 15159, 15429, 16077, 16491, 16887, 17121, 17949, 17967, 18057, 18147, 20217, 20829, 21027, 21459, 22557, 24609, 24771, 24897, 25077, 26247, 26427, 26841, 27507, 28551, 28587, 28767, 28821, 29109

Rather than sphenic numbers, with three distinct prime factors, we could consider biprimes or numbers with two distinct prime factors. Firstly let's look at numbers with properities as follows:

Biprimes containing the digit 2 whose two prime factors also contain the digit 2 and whose additive digital root is 2. 

There are 73 such numbers in the range up to 40000. The first of these is 254 = 2 * 127 with a digital root of 2. Here they are (permalink):

254, 542, 2558, 2594, 2846, 3242, 4286, 4322, 4502, 4682, 5042, 5267, 5294, 5582, 5942, 8462, 12242, 12422, 12458, 12494, 12854, 14258, 16526, 17246, 17642, 18254, 18929, 19442, 20486, 21458, 22502, 22574, 23483, 23654, 24014, 24041, 24086, 24194, 24482, 24554, 24842, 24914, 25022, 25094, 25166, 25202, 25238, 25274, 25526, 25562, 25598, 25706, 25778, 25814, 25958, 25967, 26498, 26534, 27254, 27623, 28442, 28451, 28586, 28829, 29693, 29846, 30242, 30521, 32546, 33842, 35246, 36254, 39629

Again we can consider the revised criteria:

Biprimes NOT containing the digit 2 whose two prime factors also contain the digit 2 and whose additive digital root is 2. 

There are 53 such numbers in the range up to 40000 with the first being 1046 = 2 * 523 with a digital root of 2. Here are the numbers (permalink):

1046, 1658, 3683, 4034, 4106, 4178, 4358, 4538, 4574, 4754, 4853, 4934, 5006, 5078, 5114, 5186, 5366, 5438, 5834, 5906, 6509, 6518, 7058, 7454, 7859, 10046, 11054, 11846, 13646, 14438, 14474, 15167, 15446, 16418, 17858, 19658, 30854, 31646, 33401, 34058, 34418, 34598, 35687, 35858, 36434, 36506, 36578, 37046, 37091, 37613, 38414, 38846, 39854

What's Special About Palindrome 27472?

Palindromic numbers occur every century during a millenium and the millenium I'm focused on stretches from 27000 to 27999. Because the first two digits of numbers in this millenium add to 9, the palindromes that arise have the peculiarity that the middle digit is always the arithmetic digital root, with the exception of 27072. Thus we have:

27172 with digital root of 1

27272 with digital root of 2

27372 with digital root of 3

27472 with digital root of 4

27572 with digital root of 5

27672 with digital root of 6

27772 with digital root of 7 

27872 with digital root of 8 

27972 with digital root of 9

I'll soon be 27472 days old and I've gotten into the habit of creating a post for each palindromic day. So what other special properties does this palindrome have? 

• It is a palindrome in base 9 as well $$27472_{10} \rightarrow 41614_{ \, 9}$$This qualifies it for membership in OEIS A180454: numbers that are 5-digit palindromes in at least two bases.
  
  
• It is a d-powerful number, because it can be written as $$27472=2^3 + 7^4 + 4^3 + 7^5 + 2^{13} $$
• It can be written as a sum of two squares in two different ways because it is a product of a power of 2 and two 4k+1 primes:$$ \begin{align} 27472 &= 2^4 \times 17 \times 101\\ &=24^2+ 164^2\\ &=56^2+ 156^2 \end{align}$$
•  27472 is an untouchable number, because it is not equal to the sum of proper divisors of any number.
  
  
• 27472 is a palindrome with exactly six prime factors (counted with multiplicity) and this qualifies it for membership in OEIS A046332 whose members, up to 40000, are:
  
  
  2772, 2992, 6776, 8008, 21112, 21712, 21912, 23632, 23832, 25452, 25752, 25952, 27472, 28782, 29392
  
  
• 27472 is a palindrome which is even and in which the parity of digits alternates. This qualifies it for membership in OEIS A030149. The initial members are:
  
  
  0, 2, 4, 6, 8, 212, 232, 252, 272, 292, 414, 434, 454, 474, 494, 616, 636, 656, 676, 696, 818, 838, 858, 878, 898, 21012, 21212, 21412, 21612, 21812, 23032, 23232, 23432, 23632, 23832, 25052, 25252, 25452, 25652, 25852, 27072, 27272, 27472
  
  
• The Collatz Trajectory for 27472 is:

27472, 13736, 6868, 3434, 1717, 5152, 2576, 1288, 644, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1

There are 108 steps required to reach 1. Figure 1 shows a plot of the numbers using a logarithmic scale for the vertical axis. 

 

Figure 1

• The Aliquot Sequence for 27472 is:

27472, 29444, 25240, 31640, 50440, 73040, 114448, 117680, 156112, 174224, 163366, 121862, 81418, 40712, 46648, 61352, 53698, 26852, 28210, 36302, 25954, 15086, 8794, 4400, 7132, 5356, 4836, 7708, 6404, 4810, 4766, 2386, 1196, 1156, 993, 335, 73, 1, 0

            Figure 2 shows a plot of these numbers. 

Figure 2

• The Anti-Divisors of 27472 are:

3, 5, 7, 9, 11, 15, 27, 32, 33, 37, 45, 47, 55, 99, 111, 135, 165, 167, 185, 297, 329, 333, 407, 495, 544, 555, 999, 1169, 1221, 1485, 1665, 2035, 3232, 3663, 4995, 6105, 7849, 10989, 18315 

• The Arithmetic Derivative of 27472 is 56832
  

• The Maximum - Minimum Recursive Algorithm for 27472 produces:

27472, 54945, 50985, 92961, 86922, 75933, 63954, 61974, 82962, 75933 

• The Minimal Goldbach decomposition of 27472 is 23 and 27449
  

• The number of steps required is to reach home prime is 5 :
• 27472
• 222217101
• 333310925169
• 3365956198099
• 1111271910230901
• 3419034730977487
  

• The multiplicative persistence of 27472 is as follows: 27472, 784, 224, 16, 6
  
  
• 27472 is a Zumkeller number, because its divisors can be partitioned in two sets with the same sum (28458). There are many such partitions, one example of which is:
  
  [8, 34, 136, 808, 27472] and [1, 2, 4, 16, 17, 68, 101, 202, 272, 404, 1616, 1717, 3434, 6868, 13736] both of which sum to 28458
  
  The divisors of 27472 are 1, 2, 4, 8, 16, 17, 34, 68, 101, 136, 202, 272, 404, 808, 1616, 1717, 3434, 6868, 13736 and 27472.
  

• 27472 has Odds and Evens Trajectory of length 1 and is 27472, 27478, 27478
  
  
• 27472 is a pseudoperfect number because it is the sum of a subset of its proper divisors. There are many such subsets one of which is [101, 1616, 1717, 3434, 6868, 13736]. Permalink.
  
  
• 27472 is a practical number, because each smaller number is the sum of distinct divisors of 27472.  For example, take an arbitrary number like 23891. It can expressed as the sum of distinct divisors of 24742 in several different ways e.g. the sum of 1, 17, 68, 272, 404, 808, 1717, 6868 and 13736.
  
  
• 27472 is of course an abundant number, since it is smaller than the sum of its proper divisors (29444).

27472 has the property, shared by all the numbers in its decade, that its digit sum is given by the concatenation of its first and last digit, here 22. Thus:

• 27470 has digit sum 20 
• 27471 has digit sum 21 
• 27472 has digit sum 22
• 27473 has digit sum 23 
• 27474 has digit sum 24 
• 27475 has digit sum 25 
• 27476 has digit sum 26 
• 27477 has digit sum 27 
• 27478 has digit sum 28 
• 27479 has digit sum 29

This permalink will generate a list of all 1230 numbers in the range up to 40000. However, there are only 13 palindromes with this property and they are 191, 2992, 10901, 11711, 12521, 13331, 14141, 25852, 26662, 27472, 28282, 29092 and 39993.

Under Conway's Game of Life rules, the 27472 shape shown at the beginning of this blog stabilises after about 914 generations to the shapes shown in Figures 3 and 4 with the paths of the five gliders visible in Figure 4.

Figure 3

Figure 4

More About Balanced Numbers

On Friday, the 24th March 2023 (almost a year ago now), I created a post titled Balanced Numbers. I defined these numbers to be those whose sums of digits to the left and right of their centre points or centre digits were equal. Any number with \(2k\) digits where \(k=1,2,3, \dots \) will have a centre point, for example 2314 where:$$ \overbrace{23}^{\text{sum is 5}} \, \overbrace{14}^{\text{sum is 5}}$$Any number with \(2k+1\) digits where \(k=1,2,3, \dots\) will have a centre digit, for example 27018 where$$ \overbrace{27}^{\text{sum is 9}}  0  \overbrace{18}^{\text{sum is 9}}$$There are 2764 such numbers in the range up to 40,000. However, there is a special category of balanced numbers that are far less numerous and these have the additional property that they have an odd number of digits such that the centre digit is equal to the arithmetic digital root of the number. An example is the number associated with my diurnal age today, 27363 with an arithmetic digital root of 3:$$ \overbrace{27}^{\text{sum is 9}}  \underbrace{3}_{\text{root}}   \overbrace{63}^{\text{sum is 9}}$$As would be expected, these numbers comprise about 10% of the total number of balanced numbers and in the range up to 40,000, there are 271 of them. One of them contains three digits: 999. The rest contain five digits and these are:

18109, 18118, 18127, 18136, 18145, 18154, 18163, 18172, 18181, 18190, 18209, 18218, 18227, 18236, 18245, 18254, 18263, 18272, 18281, 18290, 18309, 18318, 18327, 18336, 18345, 18354, 18363, 18372, 18381, 18390, 18409, 18418, 18427, 18436, 18445, 18454, 18463, 18472, 18481, 18490, 18509, 18518, 18527, 18536, 18545, 18554, 18563, 18572, 18581, 18590, 18609, 18618, 18627, 18636, 18645, 18654, 18663, 18672, 18681, 18690, 18709, 18718, 18727, 18736, 18745, 18754, 18763, 18772, 18781, 18790, 18809, 18818, 18827, 18836, 18845, 18854, 18863, 18872, 18881, 18890, 18909, 18918, 18927, 18936, 18945, 18954, 18963, 18972, 18981, 18990, 27109, 27118, 27127, 27136, 27145, 27154, 27163, 27172, 27181, 27190, 27209, 27218, 27227, 27236, 27245, 27254, 27263, 27272, 27281, 27290, 27309, 27318, 27327, 27336, 27345, 27354, 27363, 27372, 27381, 27390, 27409, 27418, 27427, 27436, 27445, 27454, 27463, 27472, 27481, 27490, 27509, 27518, 27527, 27536, 27545, 27554, 27563, 27572, 27581, 27590, 27609, 27618, 27627, 27636, 27645, 27654, 27663, 27672, 27681, 27690, 27709, 27718, 27727, 27736, 27745, 27754, 27763, 27772, 27781, 27790, 27809, 27818, 27827, 27836, 27845, 27854, 27863, 27872, 27881, 27890, 27909, 27918, 27927, 27936, 27945, 27954, 27963, 27972, 27981, 27990, 36109, 36118, 36127, 36136, 36145, 36154, 36163, 36172, 36181, 36190, 36209, 36218, 36227, 36236, 36245, 36254, 36263, 36272, 36281, 36290, 36309, 36318, 36327, 36336, 36345, 36354, 36363, 36372, 36381, 36390, 36409, 36418, 36427, 36436, 36445, 36454, 36463, 36472, 36481, 36490, 36509, 36518, 36527, 36536, 36545, 36554, 36563, 36572, 36581, 36590, 36609, 36618, 36627, 36636, 36645, 36654, 36663, 36672, 36681, 36690, 36709, 36718, 36727, 36736, 36745, 36754, 36763, 36772, 36781, 36790, 36809, 36818, 36827, 36836, 36845, 36854, 36863, 36872, 36881, 36890, 36909, 36918, 36927, 36936, 36945, 36954, 36963, 36972, 36981, 36990 

Notice how the first two digits and last two digits of these numbers are all multiples of 9. This would continue with the next five digit numbers being 45109, 45118, 45127 etc. This sequence is not in the OEIS and I don't intend to propose its inclusion but there is a somewhat related sequence and that is OEIS A240927:

A240927

Positive integers with \(2k\) digits (the first of which is not 0) where the sum of the first \(k\) digits equals the sum of the last \(k\) digits.

The members of this sequence are a subset of the Balanced Numbers that I wrote about in my blog post. It should be noted that my use of the term "balanced number" is peculiar to me although not exclusively so. Figure 1 shows an example of another site that uses the term in the same way:

Figure 1: source

Figure 2 shows another example from Geeks for Geeks:

Figure 2: source

Every Number Is Interesting

It's been quite some time since I last posted. My temporary relocation to the wilds of Sumatra has hampered my blogging opportunities but I wanted to make at least one post before July is over. It's July 31st now but on July 29th I turned 26780 days old. Initially, I could find little of interest about this number but I knew this was only because I wasn't looking hard enough. 

Once I did I soon realised that 26780 is special in at least one way, namely none of its digits repeat. This led me to ask how many such five digit numbers have this property? The five digit numbers range from 10000 to 99999 and there are thus 89999 of them. If there are to be five digits and none of them can repeat then the first, or leftmost digit, cannot be zero so there are 9 possibilities. For the second digit there are also 9 possibilities, followed by 8, 7 and 6 for the remaining positions, giving at total of 9 x 9 x 8 x 7 x 6 = 27216 numbers. This represents 30.24% of the range. 

Thus 26780 is special but not that special because about three out of every ten five digit numbers have this same property. Examining the number further, I noticed that the digit sum of 26780 is 23 and thus it has a digital root of five. Thus it's a five digit number with a digital root of five. Numbers with these two properties are ten times as rare. In fact there are 3024 of them giving a percentage of 3.36%. This makes the number a lot more special.

On still further investigation, I noticed that 26780 has five prime factors, counting multiplicity, because it factorises to 2 x 2 x 5 x 13 x 103. There are only  253 such numbers out of the 89999 and this represents 0.0028%, making the number something of a rara avis. Here is the list of such numbers:

10256, 10625, 10832, 12056, 12650, 13064, 13208, 13496, 14036, 14360, 14576, 14630, 14756, 14792, 15260, 15368, 15620, 15728, 15980, 16304, 16340, 16952, 17204, 17240, 17384, 17420, 17456, 17528, 17960, 18536, 18590, 19256, 19472, 19580, 19652, 19832, 20984, 23180, 23450, 23540, 23576, 23864, 24368, 25016, 25340, 25916, 25970, 26780, 27140, 27536, 27860, 27896, 27950, 28175, 28490, 28760, 28976, 29768, 29876, 30416, 30758, 30875, 31064, 31208, 31496, 31568, 31784, 31820, 31892, 31928, 32450, 32504, 34250, 34520, 34790, 34916, 34952, 35096, 35240, 36140, 36248, 36752, 37625, 37940, 38012, 38120, 38570, 39056, 39128, 39416, 39650, 39704, 39875, 40136, 40172, 40568, 40712, 41756, 41936, 42980, 43016, 43250, 43610, 45032, 45392, 46508, 46580, 46832, 47012, 47138, 47192, 47912, 48650, 48920, 49028, 49352, 49532, 49820, 50468, 50648, 50792, 51260, 51620, 51980, 52016, 52430, 53168, 53690, 53924, 54230, 54392, 54608, 54680, 54860, 56012, 56210, 56408, 56912, 56984, 57128, 57380, 58190, 58460, 58712, 59180, 59216, 59720, 59864, 60125, 60152, 60872, 61304, 62150, 62348, 62510, 62780, 63752, 63824, 64130, 65048, 65192, 65219, 65480, 67208, 67298, 67928, 68072, 68324, 68450, 68540, 69125, 69350, 69584, 70196, 71825, 71852, 72590, 72968, 73508, 73580, 73625, 73850, 74192, 75164, 75416, 75803, 76280, 76532, 76820, 79160, 79250, 79268, 79304, 79430, 80132, 80276, 80456, 80465, 81032, 81356, 81392, 81752, 81950, 82076, 82760, 82904, 83912, 84632, 85316, 85640, 85910, 86504, 86792, 87125, 87620, 89420, 89456, 89672, 89726, 90248, 90356, 90428, 91364, 91472, 91580, 91832, 91850, 92876, 93128, 93560, 93740, 94280, 94316, 94352, 94820, 94856, 95108, 95432, 95612, 95720, 95810, 95864, 96278, 96350, 96530, 96728, 97016, 97250, 97340, 97682, 98150, 98456, 98735

Such numbers might well find a place in the Online Encyclopedia of Integer Sequence (OEIS) but I'll not be proposing them as a sequence given that I don't contribute anymore. The sequence can be stated as follows:

Five digit numbers in which no digit is repeated, the digital root is 5 and there are five prime factors (not necessarily distinct)

The reasons I've given for not interacting with the OEIS are stated in an earlier post. The point of this post is to show that every number has interesting properties if you look hard enough.

ADDENDUM: October 19th 2022

I had to include the following that I discovered here as it's relevant to the content of this post. 

Folklore tells us that there are no uninteresting natural numbers. The argument hinges on the following observation: Every subset of the natural numbers is either empty, or has a smallest element. The argument usually goes something like this. If there would be any uninteresting natural numbers, the set U of all these uninteresting natural numbers would have a smallest element, say u ∈ U. But u in itself has a very remarkable property. u is the smallest uninteresting natural number, which is very interesting indeed. So U, the set of all the uninteresting natural numbers, can not have a smallest element, therefore U must be empty. In other words, all natural numbers are interesting.

Multiplicative Persistence and Multiplicative Digital Root

My diurnal age today (26748) has the property that it has a multiplicative persistence of 6. This qualifies it for membership in OEIS A199996:

A199996

Composite numbers whose multiplicative persistence is 6.

The initial members of the sequence are:

6788, 6878, 6887, 7688, 7868, 7886, 8678, 8687, 8768, 8786, 8876, 16788, 16878, 16887, 17688, 17868, 17886, 18678, 18687, 18768, 18786, 18867, 18876, 23788, 23878, 24678, 24687, 24768, 24786, 24867, 24876, 26478, 26487, 26748, 26784, 26847, 26874, 27388 ...

To quote from Wikipedia:

In number theory, the multiplicative digital root of a natural number \(n\) in a given number base \(b\) is found by multiplying the digits of \(n\) together, then repeating this operation until only a single-digit remains, which is called the multiplicative digital root of \(n\). Multiplicative digital roots are the multiplicative equivalent of digital roots.

The number of iterations required to reach the multiplicative digital root is termed the multiplicative persistence. It is conjectured that there is no number with a multiplicative persistence greater than 11. The smallest numbers with multiplicative persistence of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 are:

0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899

These numbers constitute OEIS A003001 in the OEIS (see permalink for an algorithm that will generate the members of this sequence up to one million). 

Here is permalink to an algorithm that will calculate multiplicative persistence and multiplicative digital roots for a range of numbers (both composite and prime). The algorithm is easily modified to search for a specific multiplicative persistence or multiplicative digital root. Primes can be excluded by simply adding that condition to the relevant section of the code.

ADDENDUM

On October 4th 2022, my diurnal age was 26847, a permutation of the digits of 26748 and thus also having a multiplicative persistence of 6. In between these two diurnal ages, I passed 26784 days, another permutation, and 26874 is still to come. Altogether there are 120 permutations of the digits 2, 4, 6, 7 and 8. 

It's interesting to look at a breakdown of the percentages of numbers with various multiplicative persistences. Up to one million, the breakdown is:

7    0.245%

6    0.449%

5    2.47%

4    6.68%

3    12.4%

2    37.5%

1    40.3%

In this range there are no numbers with a multiplicative persistence of 8. The first such number is 2,677,889.