Metadromes

To quote from Numbers Aplenty:

A number is a metadrome in a given base \(b\) (often 10 or 16) if its digits are in strictly increasing order in that base. For example, 1234, 68 and 12789 are all metadromes in base 10. The total number of metadromes in base \(b\)  is equal to \(2^{b-1}\), hence in base 10 there are \(2^{10-1} = 2^9=512\) metadromes ranging from 0 to 123456789.  

For some reason, I've ignored these numbers over the years even though they make their appearance regularly in Numbers Aplenty. Here is the full list of base 10 metadromes:

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 56, 57, 58, 59, 67, 68, 69, 78, 79, 89, 123, 124, 125, 126, 127, 128, 129, 134, 135, 136, 137, 138, 139, 145, 146, 147, 148, 149, 156, 157, 158, 159, 167, 168, 169, 178, 179, 189, 234, 235, 236, 237, 238, 239, 245, 246, 247, 248, 249, 256, 257, 258, 259, 267, 268, 269, 278, 279, 289, 345, 346, 347, 348, 349, 356, 357, 358, 359, 367, 368, 369, 378, 379, 389, 456, 457, 458, 459, 467, 468, 469, 478, 479, 489, 567, 568, 569, 578, 579, 589, 678, 679, 689, 789, 1234, 1235, 1236, 1237, 1238, 1239, 1245, 1246, 1247, 1248, 1249, 1256, 1257, 1258, 1259, 1267, 1268, 1269, 1278, 1279, 1289, 1345, 1346, 1347, 1348, 1349, 1356, 1357, 1358, 1359, 1367, 1368, 1369, 1378, 1379, 1389, 1456, 1457, 1458, 1459, 1467, 1468, 1469, 1478, 1479, 1489, 1567, 1568, 1569, 1578, 1579, 1589, 1678, 1679, 1689, 1789, 2345, 2346, 2347, 2348, 2349, 2356, 2357, 2358, 2359, 2367, 2368, 2369, 2378, 2379, 2389, 2456, 2457, 2458, 2459, 2467, 2468, 2469, 2478, 2479, 2489, 2567, 2568, 2569, 2578, 2579, 2589, 2678, 2679, 2689, 2789, 3456, 3457, 3458, 3459, 3467, 3468, 3469, 3478, 3479, 3489, 3567, 3568, 3569, 3578, 3579, 3589, 3678, 3679, 3689, 3789, 4567, 4568, 4569, 4578, 4579, 4589, 4678, 4679, 4689, 4789, 5678, 5679, 5689, 5789, 6789, 12345, 12346, 12347, 12348, 12349, 12356, 12357, 12358, 12359, 12367, 12368, 12369, 12378, 12379, 12389, 12456, 12457, 12458, 12459, 12467, 12468, 12469, 12478, 12479, 12489, 12567, 12568, 12569, 12578, 12579, 12589, 12678, 12679, 12689, 12789, 13456, 13457, 13458, 13459, 13467, 13468, 13469, 13478, 13479, 13489, 13567, 13568, 13569, 13578, 13579, 13589, 13678, 13679, 13689, 13789, 14567, 14568, 14569, 14578, 14579, 14589, 14678, 14679, 14689, 14789, 15678, 15679, 15689, 15789, 16789, 23456, 23457, 23458, 23459, 23467, 23468, 23469, 23478, 23479, 23489, 23567, 23568, 23569, 23578, 23579, 23589, 23678, 23679, 23689, 23789, 24567, 24568, 24569, 24578, 24579, 24589, 24678, 24679, 24689, 24789, 25678, 25679, 25689, 25789, 26789, 34567, 34568, 34569, 34578, 34579, 34589, 34678, 34679, 34689, 34789, 35678, 35679, 35689, 35789, 36789, 45678, 45679, 45689, 45789, 46789, 56789, 123456, 123457, 123458, 123459, 123467, 123468, 123469, 123478, 123479, 123489, 123567, 123568, 123569, 123578, 123579, 123589, 123678, 123679, 123689, 123789, 124567, 124568, 124569, 124578, 124579, 124589, 124678, 124679, 124689, 124789, 125678, 125679, 125689, 125789, 126789, 134567, 134568, 134569, 134578, 134579, 134589, 134678, 134679, 134689, 134789, 135678, 135679, 135689, 135789, 136789, 145678, 145679, 145689, 145789, 146789, 156789, 234567, 234568, 234569, 234578, 234579, 234589, 234678, 234679, 234689, 234789, 235678, 235679, 235689, 235789, 236789, 245678, 245679, 245689, 245789, 246789, 256789, 345678, 345679, 345689, 345789, 346789, 356789, 456789, 1234567, 1234568, 1234569, 1234578, 1234579, 1234589, 1234678, 1234679, 1234689, 1234789, 1235678, 1235679, 1235689, 1235789, 1236789, 1245678, 1245679, 1245689, 1245789, 1246789, 1256789, 1345678, 1345679, 1345689, 1345789, 1346789, 1356789, 1456789, 2345678, 2345679, 2345689, 2345789, 2346789, 2356789, 2456789, 3456789, 12345678, 12345679, 12345689, 12345789, 12346789, 12356789, 12456789, 13456789, 23456789, 123456789 

Of these 512 metadromes, 100 or about 20% are prime. \(p_{13479}=145679\) is the largest metadromic prime whose index is a metadrome too. Here are the primes:

2, 3, 5, 7, 13, 17, 19, 23, 29, 37, 47, 59, 67, 79, 89, 127, 137, 139, 149, 157, 167, 179, 239, 257, 269, 347, 349, 359, 367, 379, 389, 457, 467, 479, 569, 1237, 1249, 1259, 1279, 1289, 1367, 1459, 1489, 1567, 1579, 1789, 2347, 2357, 2389, 2459, 2467, 2579, 2689, 2789, 3457, 3467, 3469, 4567, 4679, 4789, 5689, 12347, 12379, 12457, 12479, 12569, 12589, 12689, 13457, 13469, 13567, 13679, 13789, 15679, 23459, 23567, 23689, 23789, 25679, 34589, 34679, 123457, 123479, 124567, 124679, 125789, 134789, 145679, 234589, 235679, 235789, 245789, 345679, 345689, 1234789, 1235789, 1245689, 1456789, 12356789, 23456789

Of these 512 metadromes, 131 have prime factors that are also metadromes. The largest of these is \(1235689 = 7 \times 13 \times 37 \times 367\). Here are the numbers:

1, 4, 6, 8, 9, 12, 14, 15, 16, 18, 24, 25, 26, 27, 28, 34, 35, 36, 38, 39, 45, 46, 48, 49, 56, 57, 58, 68, 69, 78, 125, 126, 128, 134, 135, 136, 138, 145, 147, 148, 156, 158, 168, 169, 178, 189, 234, 235, 236, 237, 238, 245, 247, 256, 259, 267, 268, 278, 289, 345, 348, 356, 357, 358, 368, 378, 456, 459, 468, 469, 478, 567, 578, 1235, 1239, 1246, 1248, 1256, 1258, 1269, 1345, 1357, 1368, 1369, 1456, 1458, 1468, 1479, 1568, 2345, 2346, 2349, 2368, 2457, 2478, 2479, 2569, 2578, 2679, 3456, 3458, 3468, 3478, 3578, 5678, 12348, 12358, 12467, 12478, 12789, 13456, 13467, 13468, 13579, 13689, 15678, 23569, 24589, 24678, 24679, 25678, 34568, 34569, 124579, 124689, 134568, 134589, 134689, 234567, 234689, 1235689

There are only 30 numbers that are metadromes in base 8 and base 10 and these are (permalink):

1, 2, 3, 4, 5, 6, 7, 12, 13, 14, 15, 19, 23, 28, 29, 37, 38, 39, 46, 47, 156, 157, 158, 159, 167, 238, 239, 247, 678, 679

See Table 1 for the conversions to base 8. There are of course only \(2^{8-1} = 2^7 =128 \) metadromes in base 8. 

Table 1: permalink

The algorithm that the permalink refers to can be used to generate similar tables for bases 9, 7, 6, 5, 4, 3 and 2, although for 2 there is only the number 1 that satisfies. The algorithm will not work for bases greater than 10, although it would be interesting to try to develop one.

Properties of Concatenated Numbers

Let's split a number in two parts and then consider the relationship that these two parts bear to the number of which both are a part. Now I've done this before as described in a post titled Energetic Numbers. In this post, I looked at numbers that belong to OEIS  A055480:

A055480

Energetic numbers: numbers that can be broken into two or more substrings and expressed as a sum of (possibly different) positive powers of those substrings.

This time I'll be looking at two different types of relationships. Let's consider 27573 and break it into two parts: 275 and 73. It turns out that my diurnal age today (27573) has an interesting property and it involves the totient function, commonly referred to as Euler's phi function or simply the phi function. We find that:$$ \begin{align}  \phi(27573) &= 1440 \\ \phi(275) \times \phi(73) &= 200 \times 72\\ &=1440 \end{align} $$Numbers like this belong to OEIS A147619 :

 A147619             Numbers \(n\)  = \(a \,| \, b\) such that \( \phi(n) = \phi(a) \times \phi(b) \)
                                where the symbol | represents concatenation

The initial members of this sequence are:

78, 780, 897, 918, 1179, 1365, 1776, 2574, 2598, 2967, 3168, 3762, 4758, 5775, 5796, 7800, 7875, 7917, 8217, 8970, 9180, 9576, 11790, 13650, 13662, 13875, 13896, 14391, 17760, 18564, 18858, 19812, 20097, 25740, 25935, 25974, 25980, 27573, 28776, 28779, 29670, 31680, 33165, 35919, 37620

Let's look at the breakdown for the numbers between 27573 and 37620:

• 275 | 73  -->  27573
  
  
•  287 | 76  -->  28776
  
  
•  287 |  79  -->  28779
  
  
•  29 | 670  -->   29670
  
  
•  31 | 680  -->   31680
  
  
•  331 | 65  -->   33165
  
  
•  35 | 919  -->   35919
  
  
•  37 | 620  -->   37620

A similar thing can be done with the sigma function that tallies the sum of the divisors of a number. Once again we break the number into two parts and then use the sigma function to connect those two parts to the whole. There is no OEIS function this time around but the property of the numbers being considered is as follows:

 Numbers \(n\)  = \(a \,| \, b\) such that \( \sigma(n) = \sigma(a) \times \sigma(b) \)
where the symbol | represent concatenation

The 78 numbers satisfying this condition between 27000 and 40000 are (permalink):

27175, 27298, 27418, 27445, 27500, 28018, 28195, 28750, 28798, 28978, 29038, 29058, 29098, 29278, 29395, 29398, 29500, 29875, 29875, 29950, 29950, 29980, 30498, 30775, 30788, 30989, 31750, 31795, 31918, 32595, 33175, 33238, 33298, 33725, 34189, 34555, 34557, 34795, 34975, 35338, 35395, 35578, 35585, 35670, 35818, 35975, 36178, 36238, 36350, 36398, 36426, 36775, 37138, 37241, 37678, 37750, 37798, 38247, 38370, 38518, 38570, 38638, 38750, 38826, 38856, 38877, 38940, 38975, 39118, 39425, 39478, 39500, 39754, 39805, 39898, 39976, 39980, 39992

Table 1 shows the breakdown for numbers between 27000 and 30000:

Table 1

Let's look at the first number 27175 where we break the number into 271 and 75 and we have:$$ \begin{align} \sigma(27175) &= 33728\\ \sigma(271) \times \sigma(75) &= 272 \times 124\\ &= 33728 \end{align}$$There are other possible concatenations involving more than two parts or other functions and properties of numbers so this post is merely a sample.

Palindromic Day 27572

Every 100 days another palindrome day rolls by and yesterday I celebrated palindromic day 27573. 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.

Digit Permutations of Numbers

I've written about permutations of the digits of a number in many posts. For example, in my recent post titled Determinants of Circulant Matrices (11th of August 2024) I looked at circulant matrices that have determinants with digits that are permutations of the number's digits. In this current post, I'll be looking at some new types, specifically where the digits of the number are equal to:

• a permutation of the digits of the aliquot sum of the divisors of the number
• a permutation of the digits of the sum of the divisors of the number
• a permutation of the digits of the totient of the number

SUM OF PROPER DIVISORS

There are 37 numbers in the range up to 40,000 that have the same digits as the sum of their proper divisors. These numbers are listed below together with the permutation. Notice that perfect numbers remain unchanged. Permalink.

(6, 6), (28, 28), (411, 141), (496, 496), (604, 460), (1305, 1035), (3664, 3466), (4086, 4806), (4672, 4726), (4896, 9846), (5046, 5406), (7785, 5787), (8128, 8128), (8739, 3897), (9331, 1933), (14535, 13545), (16012, 12016), (18342, 21438), (18585, 18855), (19648, 19468), (20634, 23046), (21534, 23154), (21628, 16228), (22365, 22563), (25911, 11529), (27568, 25876), (28108, 21088), (29160, 69210), (29188, 21898), (31185, 38511), (32091, 13029), (32271, 12273), (34956, 53496), (35898, 38598), (35925, 23595), (36172, 27136), (37698, 39678)

These numbers constitute OEIS A085844 and they are:

6, 28, 411, 496, 604, 1305, 3664, 4086, 4672, 4896, 5046, 7785, 8128, 8739, 9331, 14535, 16012, 18342, 18585, 19648, 20634, 21534, 21628, 22365, 25911, 27568, 28108, 29160, 29188, 31185, 32091, 32271, 34956, 35898, 35925, 36172, 37698

SUM OF DIVISORS

There are 45 numbers in the range up to 40,000 that have the same digits as the sum of their divisors including the number itself this time. These numbers are listed below together with the permutations. Permalink.

(1, 1), (69, 96), (258, 528), (270, 720), (276, 672), (609, 960), (639, 936), (2391, 3192), (2556, 6552), (2931, 3912), (3409, 3904), (3678, 7368), (3679, 3976), (4291, 4912), (5092, 9520), (6937, 7936), (8251, 8512), (10231, 11032), (12087, 18720), (12931, 13192), (15480, 51480), (16387, 18736), (20850, 52080), (22644, 62244), (22893, 32928), (24369, 32496), (26145, 52416), (26442, 62244), (27846, 78624), (28764, 78624), (29880, 98280), (29958, 59928), (30823, 33208), (31812, 81312), (32658, 65328), (34207, 34720), (34758, 75348), (34909, 39904), (36045, 65340), (36249, 49632), (36729, 67392), (36978, 73968), (36990, 99360), (38491, 39184), (38538, 83538)

These numbers constitute OEIS A115920 and they are:

1, 69, 258, 270, 276, 609, 639, 2391, 2556, 2931, 3409, 3678, 3679, 4291, 5092, 6937, 8251, 10231, 12087, 12931, 15480, 16387, 20850, 22644, 22893, 24369, 26145, 26442, 27846, 28764, 29880, 29958, 30823, 31812, 32658, 34207, 34758, 34909, 36045, 36249, 36729, 36978, 36990, 38491, 38538

TOTIENT

There are 36 numbers in the range up to 40,000 that have the same digits as their totients. These numbers are listed below together with the permutations. Permalink.

(1, 1), (21, 12), (63, 36), (291, 192), (502, 250), (2518, 1258), (2817, 1872), (2991, 1992), (4435, 3544), (5229, 2952), (5367, 3576), (5637, 3756), (6102, 2016), (6174, 1764), (6543, 4356), (6822, 2268), (7236, 2376), (7422, 2472), (8022, 2280), (8541, 5184), (8982, 2988), (17631, 11736), (18231, 11832), (18261, 12168), (20301, 13200), (20518, 10258), (20617, 20176), (21058, 10528), (22471, 21472), (22851, 15228), (25196, 12596), (25918, 12958), (27615, 12576), (29817, 19872), (34816, 16384), (35683, 33568)

These numbers constitute OEIS A115921 and all except 1 are deficient. They are:

1, 21, 63, 291, 502, 2518, 2817, 2991, 4435, 5229, 5367, 5637, 6102, 6174, 6543, 6822, 7236, 7422, 8022, 8541, 8982, 17631, 18231, 18261, 20301, 20518, 20617, 21058, 22471, 22851, 25196, 25918, 27615, 29817, 34816, 35683

More Numbers Within Numbers

In December of 2022, I made a post titled Numbers Within Numbers and so for this post I've made the title More Numbers Within Numbers but the types of numbers considered in the former post are quite different to the ones I'll be considering in this post. The idea for this post came from a peculiarity in the number associated with my diurnal age today: 27567.

What I mean by a number within a number in this present context is simply a substring of the number viewed as a string. Here the substring being considered is "27" that is contained within the larger string "27567". Thus we see that:$$ {\Large \textbf{27} 567}$$Now let's consider the sum of digits of the number:$${\Large \textbf{27} 567 \rightarrow  2 + 7 + 5 + 6+7= \textbf{27}}$$Let's move on to the factorisation where we have:$${\Large \textbf{27} 567= \textbf{27} \times 1021}$$Now what about the totient? We find that 27567 has a totient of 18360 and$$ {\Large 18360 = 5 \times 8 \times 17 \times \textbf{27}} $$Lastly, let's find the absolute value of the determinant of the circulant matrix of 27567. It turns out to be 6777 and$$  {\Large 6777  = \textbf{27} \times 251}  $$Thus in the case of 27567, the number within a number (27) turns up:

• in the digits of the number
• as the sum of the digits of the number
• in the factors of the number
• in the factors of the totient
• in the factors of the determinant of the circulant matrix

It can be noted that 27567 is a Harshad number since it is a multiple of its sum of digits (27), and also a Moran number because the ratio is a prime number: 1021 = 27567 / (2 + 7 + 5 + 6 + 7).

The natural question to ask then is how many numbers in the range up to 40000 have this property? It turns out that there are only 18 such numbers with details as shown in Table 1.

Table 1: permalink

So what about other numbers? Let's start with a substring "1". There are five numbers satisfying the previous criteria in the range up to 40000. See Table 2.

Table 2: permalink

For substring "2", there are four numbers satisfying the criteria. See Table 3.

Table 3: permalink

For the substring "3", there are no numbers that meet the criteria. For the substring "4", there are appropriately four numbers that satisfy. See Table 4.

Table 4: permalink

For the substring "5", there are three numbers that meet the criteria. See Table 5.

Table 5: permalink

For substrings "6" and "7", no numbers qualify but for the substring "8" there are three numbers that do. See Table 6.

Table 6: permalink

There are no numbers that satisfy for "9" but there are 23 numbers that satisfy for "10". See Table 7.

Table 7: permalink

For "11", no numbers that satisfy but for "12" there are 39 numbers that satisfy. See Table 8.

Table 8: permalink

For "13", there are two numbers that satisfy. See Table 9.

Table 9: permalink

For "14", there are three numbers that satisfy. See Table 10.

Table 10: permalink

For "15", there are 26 numbers that satisfy. See Table 11.

Table 11: permalink

For "16", there are 14 numbers that satisfy. See Table 12.

Table 12: permalink

For "17", there is only one number. See Table 13.

Table 13: permalink

For "18", there is a grand total of 69 numbers that qualify. See Table 14.

Table 14: permalink

For "19", there are no numbers that qualify but for "20" there are four. See Table 15.

Table 15: permalink

For "21", there are three numbers that qualify. See Table 16.

Table 16: permalink

For "22", there are two numbers that satisfy. See Table 17.

Table 17: permalink

For "23", there is only one number that satisfies. See Table 18.

Table 18: permalink

For "24", there are nine numbers that satisfy. See Table 19.

Table 19: permalink

For "25", there are four numbers that satisfy. See Table 20.

Table 20: permalink

For "26", there are no numbers that qualify and we've already dealt with "27". For "28", there is only one number that satisfies. See Table 21.

Table 21: permalink

That's it for particles in the range up to 40,000 as the SOD criterion makes it difficult  for the digit sum to reach these higher particles. Just to illustrate with an example. Take the particle "29". If we extend the range to one million, then instead of the zero for the range up to 40,000, we have nine numbers that satisfy the criteria. See Table 21.

Table 21: permalink

So, an interesting exercise but purely confined to the realm of recreational Mathematics. There no real reason to conflate the digit sum of a number with its factors as well as its totient and the determinant of its circulant matrix. 

Aesthetic Numbers

An Aesthetic or Esthetic Number is defined by Numbers Aplenty as a number \(n\)  such that the difference between adjacent digits is 1 when the number is written in base \(b\). Such numbers attracted my attention because 27564, my diurnal age yesterday, is an aesthetic number in base 8 or simply 8-aesthetic because:$$27564_{10}=65654_8$$Let's look firstly at aesthetic numbers in base 10 between 10 and 40000. There are 149 such numbers and they are (permalink):

10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 101, 121, 123, 210, 212, 232, 234, 321, 323, 343, 345, 432, 434, 454, 456, 543, 545, 565, 567, 654, 656, 676, 678, 765, 767, 787, 789, 876, 878, 898, 987, 989, 1010, 1012, 1210, 1212, 1232, 1234, 2101, 2121, 2123, 2321, 2323, 2343, 2345, 3210, 3212, 3232, 3234, 3432, 3434, 3454, 3456, 4321, 4323, 4343, 4345, 4543, 4545, 4565, 4567, 5432, 5434, 5454, 5456, 5654, 5656, 5676, 5678, 6543, 6545, 6565, 6567, 6765, 6767, 6787, 6789, 7654, 7656, 7676, 7678, 7876, 7878, 7898, 8765, 8767, 8787, 8789, 8987, 8989, 9876, 9878, 9898, 10101, 10121, 10123, 12101, 12121, 12123, 12321, 12323, 12343, 12345, 21010, 21012, 21210, 21212, 21232, 21234, 23210, 23212, 23232, 23234, 23432, 23434, 23454, 23456, 32101, 32121, 32123, 32321, 32323, 32343, 32345, 34321, 34323, 34343, 34345, 34543, 34545, 34565, 34567

Here are the frequencies for base 10 numbers being aesthetic in smaller bases in the range from 10 to 40000:

• base 9 - 165 e.g. 37825 --> 56787
• base 8 - 171 e.g. 33436 --> 101234
• base 7 - 164 e.g. 36881 --> 212345
• base 6 - 188 e.g. 38657 --> 454545
• base 5 - 182 e.g. 36119 --> 2123434
• base 4 - 188 e.g. 39867 --> 21232323
• base 3 - 134 e.g. 36905 --> 1212121212
• base 2 - 12 e.g. 21845 --> 101010101010101

There are 17, 32, 61, 116, 222, 424, 813, 1556, and 2986 10-aesthetic numbers with 2, 3, 4, 5, 6, 7, 8, 9 and 10 digits respectively. Table 1 shows a screenshot of the Numbers Aplenty website featuring an interesting formula:

Table 1

Some Mathematical Induction

Here is a problem similar to one was posed in a YouTube video that I watched recently:

Prove that \(a^n-b^n\) is divisible by \(a-b\) for all positive integer values of \(n\) with \(a\) and \(b\) both positive integers and \(a \gt b\).

We can do this using mathematical induction as follows:

It's clearly true when \(n=1\) and so let's assume it's true for some value of \(k \gt 1\). This means that:$$ \frac{a^k-b^k}{a-b}=m$$where \(m\) is an integer. Can we now show that it's true for \(k+1\). Let's begin:$$ \begin{align} \frac{a^{k+1}-b^{k+1}}{a-b} &= \frac{a \, . b^k-b \, . b^k}{a-b} \\ a-b &= c \text{ where c is some positive integer} \\ a & = b+c \\ \frac{a \, . b^{k}-b \, . b^{k}}{a-b} &= \frac{(b+c) \, . b^{k}-b \, . b^{k}}{c} \\ &= \frac{b \, . b^{k} +c \,.  b^{k} - b \, . b^{k}}{c} \\ &= \frac{c \, . b^{k} }{c} \\ &=b^{k} \text{ which is an integer} \end{align} $$Since it is true for \(n=1\) and \(n=k+1\) when it is true for \(n=k\) then it is true for all \(n\).

In the video that I referred to at the start of this post, a specific instance of the more general situation was looked at, namely:$$ \text{prove that } \frac{11^n-4^n}{7} \text{ is always an integer}$$

A Special Class of Semiprimes

A well-known factorisation involves the sum of two cubes:$$ a^3 + b^3=(a+b)(a^2-ab+b^2)$$Assuming that both \(a\) and \(b\) are positive integers, then this sum of two cubes is a semiprime if \(a+b\) and \(a^2-ab+b^2\) are both prime. It's interesting to explore when this happens.

Let's begin with \(b=1\) and restrict ourselves to semiprimes between 4 and 4000. There are only seven that qualify and they are:$$9, 65, 217, 4097, 5833, 10649, 21953 $$These numbers form the initial terms of OEIS A237040. Their factorisations are as follows:$$ \begin{align} 9 &= 3^2 = 2^3 + 1^3\\65 &= 5 \times 13 = 4^3 + 1^3\\217 &= 7 \times 31 = 6^3 + 1^3\\4097 &= 17 \times 241 = 16^3 + 1^3\\5833 &= 19 \times 307 = 18^3 + 1^3\\10649 &= 23 \times 463 = 22^3 + 1^3\\21953 &= 29 \times 757 = 28^3 + 1^3 \end{align}$$The next is \(b=8\) which yields six numbers:$$35, 133, 737, 1339, 3383, 24397$$These numbers factorise as follows:$$ \begin{align} 35 &= 5 \times 7 = 3^3 + 2^3\\133 &= 7 \times 19 = 5^3 + 2^3\\737 &= 11 \times 67 = 9^3 + 2^3\\1339 &= 13 \times 103 = 11 ^3 + 2^3\\3383 &= 17 \times 199 = 15^3 + 2^3\\24397 &= 31 \times 787 = 29^3 + 2^3 \end{align} $$These terms are not listed in the OEIS database. Next we'll let \(b=27\). This yields eight numbers and we see that 35 makes a reappearance as the 2 and 3 swap places. The numbers are:$$35, 91, 1027, 2771, 8027, 17603, 21979, 39331 $$These numbers are not listed in the OEIS database and their factorisations are:$$ \begin{align} 35 &= 5 \times 7 = 2^3 + 3^3\\91 &= 7 \times 13 = 4^3 + 3^3\\1027 &= 13 \times 79 = 10^3 + 3^3\\2771 &= 17 \times 163 = 14^3 + 3^3\\8027 &= 23 \times 349 = 20^3 + 3^3\\17603 &= 29 \times 607 = 26^3 + 3^3\\21979 &= 31 \times 709 = 28^3 + 3^3\\39331 &= 37 \times 1063 = 34^3 + 3^3 \end{align} $$Here is a permalink to an algorithm that allows further exploration using additional values of \(b\). A further avenue for investigation would be to consider semiprimes that are a difference of two cubes since we know that:$$ a^3 - b^3=(a-b)(a^2+ab+b^2)$$Using \(b=-1\), we find that the following semiprimes qualify:$$26, 215, 511, 1727, 2743, 7999, 13823$$These numbers form the initial members of OEIS A242262. Their factorisations are as follows:$$ \begin{align} 26 &= 2 \times 13 = 3^3 -1^3\\215 &= 5 \times 43 = 6^3 - 1^3\\511 &= 7 \times 73 = 8^3 - 1^3\\1727 &= 11 \times 157 = 12^3 - 1^3\\2743 &= 13 \times 211 = 14^3 - 1^3\\7999 &= 19 \times 421 = 20^3-1^3\\13823 &= 23 \times 601 = 24^3-1^3 \end{align} $$

Zeisel Numbers

Digalo con numeros
Say it with Numbers

Investigating the properties of the number associated with my diurnal age yesterday (27559), I noticed that Numbers Aplenty mentions that this number is a Zeisel number with parameters (4, 3). Now this was a type of number that I'd not heard of before. Clicking on the link in previously mentioned resource, I was given this explanation:

Let us define a sequence as:$$\begin{array}{l} p_0 = 1\\ p_n = a\cdot p_{n-1}+b\ \end{array}$$where \(a,b\in\mathbb{Z}\). If the numbers \(p_1,p_2,\dots,p_k\) are all distinct primes and \(k\ge 3\), then their product is a Zeisel number.

So applied to 27599, we start with 1 to form a new number thus \(4 \times 1 + 3 = 7\). This is prime so it is used as our new input to form the next number which is \(4 \times 7 + 3 = 31\). This is also prime so we proceed using 31 as the new input. This generates \(4 \times 31 + 3 = 127\) which is prime and this sequence of prime numbers (\(7, 31,127\)) can thus form a Zeisel number: $$7 \times 31 \times 127 = 27559$$The input 127 does not produce a prime number and so no further Zeisel numbers can be formed using (4, 3) as parameters. Interestingly, the primes 7, 31 and 127 are also consecutive Mersenne primes.

The example is given in the link of \(1419 = 3 \times 11 \times 43\) that is a Zeisel number with parameters of \(a=4\) and \(b=-1\), because \(3 = 4 \times 1-1\), \(11=4 \times 3-1\) and \(43=4 \times 11-1\).

The smallest Zeisel numbers which are the product of 3, 4, 5 and 6 factors are shown in Table 1.

Table 1: link

The first Zeisel numbers are 105, 1419, 1729, 1885, 4505, 5719, 15387, 24211, 25085, 27559, 31929, 54205, 59081, 114985, 207177 (OEIS A051015). Clearly they are few and far between as the previous such number in my diurnal age count was 25085. This was before I began keeping records in my first AirTable database. After 27599, the next is 31929, a long way off.

Compressing a Hollow Cube

Rewatching the movie "Transformers", my attention was caught by the giant (presumably hollow) cube made up out of many tiny "unit" cubes, that reassembled itself into a much smaller solid cube. This got me thinking about what volume a hollow cube could be compressed into to form a solid cube. Let's consider a  hollow cube with a side of \(n\) units. It has a volume of \(n^3\) cubic units with a hollow centre of \( (n-2)^3 \) cubic units . Thus the number of unit cubes are:$$\begin{align} n^3 - (n-2)^3 &= 2 (n^2 + n(n-2) + (n-2)^2) \\ &= 2(n^2+n^2-2n+n^2-4n+4)\\ &=2(3n^2-6n+4) \end{align}$$Using this formula we can determine the size of the solid cube that can be formed from the unit cubes and how many blocks are left over from this process. See Table 1.

Table 1: permalink

From Table 1 we can see that 216 of the 218 unit cubes that comprise a hollow cube of side 7 units can be used to form a solid cube of side 6 units with only two cubes left over. See Table 2.

Table 2

Also we can see that 13824 of the 13826 unit cubes that comprise a hollow cube of side 49 units can be used to form a solid cube of side 24 units with only two cubes left over. See Table 3.

Table 3

Looking beyond the figures in Table 1, we find that hollow cubes with sides of 163, 385, 751 and 1297 units also compress to solid cubes with sides of 54, 96, 150 and 216 units with two cubes left over (permalink).

We can summarise the results for the "2 left over hollow to solid cubes" as follows:

(7, 6) gives 62.97 % ratio of volumes (solid to hollow)
(49, 24) gives 11.75 % ratio of volumes (solid to hollow)
(163, 54) gives 3.636 % ratio of volumes (solid to hollow)
(385, 96) gives 1.550 % ratio of volumes (solid to hollow)
(751, 150) gives 0.7968 % ratio of volumes (solid to hollow)
(1297, 216) gives 0.4619 % ratio of volumes (solid to hollow)

Naturally as the side length of the hollow cube increases, the ratio of the volume of the solid cube to its hollow counterpart decreases rapidly as more and more empty space is formed inside. For example, in the case of the 751 hollow cube, the ratio of the solid to hollow volumes in less than 1%. It can be noted that none of the sides of the "2 left over hollow to solid cubes" have any factors in common.

Let's see if there's a pattern in the sides of the hollow and solid cubes:

• 7 = 7 and 6 = 2 * 3
• 49 = 7^2 and 24 = 2^3 * 3
• 163 = 163 and 54 = 2 * 3^3
• 385 = 5 * 7 * 11 and 96 = 2^5 * 3
• 751 = 751 and 150 = 2 * 3 * 5^2
• 1297 = 1297 and 216 = 2^3 * 3^3

It can be seen that the solid cubes all contain "6" as a factor. Let's investigate further. Is it possible that a hollow cube with integer sides can be collapsed into a solid cube of integer side \(x\) with no unit cubes left over? If it were possible then we would have:$$ \begin{align} 2(3n^2-6n+4) &= x^3\\6n^2-12n+8 &= x^3\\x &= (6n^2-12n+8)^{1/3}  \end{align}$$Now testing up to \(n=10000\), the expression on the LHS of the equation above never produces a whole number. I'll test for larger values of \(n\) later.

What about when there are two unit cubes left over. In that case we have:$$ \begin{align} 2(3n^2-6n+4) &= x^3+2\\6n^2-12n+6 &= x^3 \\ x &= (6n^2-12n+6)^{1/3} \end{align} $$Now in this case, up to \(n=10000\), we get all the solutions shown above plus some more. These are shown in Table 4.

Table 4: permalink

So the key equation to work with is \(6n^2-12n+(8-a) \) where \(a\) is the number of unit cubes left over. Up to 10000, there are no solutions for \(a=0\) and \(a=1\) but with \(a=2\) we get the solutions shown in Table 4. Going back to Table 1 and using the values of \(a\) shown there in the column "blocks wasted" will produce other tables of solutions as shown in Table 4. There's more to be discovered here but this post is at least a start.