Circulant Matrix to the Rescue

I was stuck for several days on a number associated with my diurnal age: 27164. I couldn't anything of interest about this number, or at least nothing that interested me. In the end, I remembered the circulant matrix that is associated with each number. For the case of 27164 this matrix is shown below. $$ \begin{pmatrix}2&7&1&6&4\\4&2&7&1&6\\6&4&2&7&1\\1&6&4&2&7\\7&1&6&4&2 \end{pmatrix}$$The determinant of this matrix is 100 and there are only 89 numbers in the range up to one million that have 100 as their determinant. These numbers are:

799, 979, 997, 12674, 14762, 16427, 17246, 21476, 23564, 24617, 24653, 25436, 26345, 26741, 27164, 32465, 33455, 33554, 34445, 34454, 34526, 34535, 34544, 34553, 35345, 35354, 35435, 35444, 35543, 35642, 36254, 41267, 42356, 42716, 43355, 43445, 43454, 43535, 43544, 43625, 44345, 44354, 44435, 44453, 44534, 44543, 45263, 45344, 45353, 45434, 45443, 45533, 46172, 46532, 47621, 52634, 53246, 53345, 53444, 53453, 53534, 53543, 54335, 54344, 54353, 54362, 54434, 54443, 55334, 55433, 56423, 61724, 62147, 62543, 63452, 64235, 64271, 65324, 67412, 71642, 72461, 74126, 76214, 303040, 304030, 403030, 889898, 988898, 989888

Of these, many are permutations of the digits of 27164. These permutations are shown below in bold with 27164 itself marked in red.

12467, 12476, 12647, 12674, 12746, 12764, 14267, 14276, 14627, 14672, 14726, 14762, 16247, 16274, 16427, 16472, 16724, 16742, 17246, 17264, 17426, 17462, 17624, 17642, 21467, 21476, 21647, 21674, 21746, 21764, 24167, 24176, 24617, 24671, 24716, 24761, 26147, 26174, 26417, 26471, 26714, 26741, 27146, 27164, 27416, 27461, 27614, 27641, 41267, 41276, 41627, 41672, 41726, 41762, 42167, 42176, 42617, 42671, 42716, 42761, 46127, 46172, 46217, 46271, 46712, 46721, 47126, 47162, 47216, 47261, 47612, 47621, 61247, 61274, 61427, 61472, 61724, 61742, 62147, 62174, 62417, 62471, 62714, 62741, 64127, 64172, 64217, 64271, 64712, 64721, 67124, 67142, 67214, 67241, 67412, 67421, 71246, 71264, 71426, 71462, 71624, 71642, 72146, 72164, 72416, 72461, 72614, 72641, 74126, 74162, 74216, 74261, 74612, 74621, 76124, 76142, 76214, 76241, 76412, 76421

Admittedly, focusing on a determinant of 100 is rather arbitrary, but it is a nice round number and it certainly came to my aid in finding an interesting property of 27164. This prompted me to investigate the range of values for the determinants in the natural numbers up to 40,000. It turns out the minimum value of -19683 occurs with 9909 and the maximum value of 205821 occurs with 30990 and 39009. Figure 1 shows a plot of the values of the determinants.

Figure 1: permalink

As can be seen it is four digit numbers from 1000 up to 9999 that fluctuate between positive and negative. The three digit numbers and the five digit numbers up to 40000 are all positive. The positive/negative fluctuations recur once the six digit numbers are reached. This is to be expected of course given the way the determinant is calculated. Figure 2 a close up of the range from 10 to 10000.

Figure 2: permalink

Finding Fibonacci

The Fibonacci numbers are few and far between. Up to a little over two million, the Fibonacci numbers are:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309

However, we can find Fibonacci numbers in all sorts of places. For example, I recently turned 27034 days old and this number is a member of OEIS A272412:

A272412

Numbers \(n\) such that \( \sigma_1(n)\) is a Fibonacci number.

It so happens that \( \sigma_1(27034) = 46368 \) which is a Fibonacci number. There are only 41 such numbers in the range up to one million. They are (permalink):

1, 2, 7, 9, 66, 70, 94, 115, 119, 2479, 18084, 19180, 19290, 22060, 23156, 23178, 24934, 24956, 25756, 26715, 27034, 28678, 28965, 29578, 30094, 32253, 32793, 34113, 35365, 38635, 39319, 40963, 42493, 44413, 45223, 45653, 322032, 429424, 503175, 624027, 670975

The sum of the aliquot parts of a number is the sum of its proper divisors and so Fibonacci numbers will show up here as well. We have to exclude prime numbers in our search because their only proper divisor is 1 and so they would need to be included. It turns out that there are 175 composite numbers up to one million whose sum of proper divisors are a Fibonacci number. They are:

1, 4, 10, 18, 27, 35, 36, 49, 51, 62, 90, 91, 171, 329, 415, 473, 533, 629, 687, 713, 902, 1119, 1135, 1207, 1214, 1605, 1711, 1927, 2936, 2949, 3436, 6083, 6103, 6845, 7831, 8119, 9487, 10063, 10207, 12367, 12531, 13231, 17069, 18373, 18703, 20283, 20579, 24319, 26843, 28783, 29719, 32743, 33823, 35263, 45443, 53121, 57683, 61573, 66779, 71653, 72803, 80785, 81779, 90949, 95593, 95611, 99937, 109093, 111179, 130153, 134149, 145403, 153779, 156613, 159323, 162083, 167579, 169699, 173353, 194251, 196393, 199883, 200543, 208723, 210649, 215603, 218731, 225923, 227173, 228649, 230053, 233579, 235993, 238643, 240133, 242149, 242495, 243013, 246179, 275603, 287617, 306179, 313043, 325726, 346415, 356963, 364099, 365363, 372359, 378646, 381779, 395723, 401579, 405443, 408883, 411979, 424283, 433403, 435811, 444083, 451043, 456179, 459179, 461243, 464579, 485483, 488443, 503579, 510779, 512749, 525119, 527243, 530419, 535043, 540083, 547403, 549779, 553283, 558815, 573803, 578723, 581579, 587963, 592283, 597203, 602579, 604763, 612779, 617483, 619459, 622163, 628883, 630563, 632579, 633323, 633779, 635123, 635963, 636179, 636683, 646840, 649869, 670171, 686083, 693211, 716179, 724429, 761899, 825143, 830183, 842899, 919651, 935821, 975143, 986179

Take 51 as an example. It's proper divisors are 1, 3 and 17. These add to 21 which is a Fibonacci number. There is no associated OEIS sequence for these numbers.

Let's look at the totients of numbers. The totient of a number \(n\) is a count of how many numbers \(1 \leq k \leq n \) have the property that \( \text{gcd}(n,k)=1\) where gcd stands for greatest common divisor. The totient of 6 is 2 because 1 and 5 have this property. These numbers form OEIS A280592: 

A280592

Numbers \(n\)  such that \( \phi(n)\) is a Fibonacci number.

 Here is the list of the 134 sequence members up to one million.

1, 2, 3, 4, 6, 15, 16, 20, 24, 30, 185, 219, 273, 285, 292, 296, 304, 315, 364, 370, 380, 432, 438, 444, 456, 468, 504, 540, 546, 570, 630, 3235, 5176, 6470, 7764, 46843, 47423, 47693, 48053, 50431, 52403, 56231, 57965, 59555, 62855, 67655, 67865, 70735, 72123, 72297, 73473, 75387, 77691, 78819, 81207, 84651, 85869, 86985, 89535, 89655, 89817, 90945, 92744, 93686, 94846, 95288, 95386, 95504, 95632, 96106, 96164, 97964, 100516, 100568, 100862, 101535, 102165, 103588, 103635, 104806, 105092, 108248, 108304, 108584, 108976, 112462, 112868, 113176, 115930, 119110, 119380, 119540, 119756, 125710, 135310, 135380, 135730, 136220, 139116, 139176, 139212, 139248, 141470, 142932, 143256, 143448, 144246, 144594, 145116, 145512, 146946, 147204, 150774, 150852, 155382, 157638, 162372, 162414, 162456, 162876, 163464, 165816, 169302, 169764, 171738, 173970, 174060, 179070, 179310, 179634, 181890, 203070, 204330, 207270

Why are there no numbers from 207271 up to one million that are members of the sequence? If we extend the range to two million, there are some additional members, namely 1040075, 1304859, 1372899, 1739812 and 1830532.

Of course, we don't need to confine ourselves to the Fibonacci numbers. We could consider the Lucas numbers instead which begin with 2, 1 rather than 0, 1 like the Fibonacci. The initial Lucas numbers are:

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196

Up to one million, there are only nine numbers that qualify and they are 1, 2, 3, 4, 10, 17, 688, 1075 and 103681.

If we consider the sum of the aliquot parts however, we get 151 in the range up to one million. These are:

4, 8, 9, 21, 48, 72, 92, 115, 129, 146, 165, 187, 205, 289, 493, 965, 999, 1143, 1337, 1417, 1495, 1749, 1957, 2517, 2527, 2722, 3077, 3397, 3401, 5177, 5599, 6437, 6609, 7097, 8201, 8357, 8551, 8777, 9017, 9485, 9701, 9797, 10777, 14239, 15637, 17549, 19639, 24751, 25141, 27199, 31879, 37499, 38359, 38825, 39149, 42319, 46241, 46715, 48946, 50959, 52471, 53627, 53851, 55505, 56137, 56693, 58951, 60031, 65387, 66511, 67159, 67519, 67591, 75605, 76117, 79897, 81581, 102689, 102707, 104341, 109709, 109869, 114109, 119641, 127957, 130721, 133141, 138037, 140377, 144689, 144841, 151661, 154741, 175477, 177097, 186401, 207149, 224593, 248429, 270251, 275789, 283453, 287033, 340513, 344507, 350369, 357101, 362833, 370541, 377249, 390641, 449233, 459709, 470321, 486476, 500893, 535841, 555341, 560509, 577197, 598813, 606989, 613409, 621827, 648569, 658667, 663073, 664481, 670829, 698209, 704737, 708749, 746381, 748753, 758861, 796109, 798869, 802289, 826489, 833069, 851441, 863773, 869041, 869741, 887969, 894029, 931453, 950353, 956509, 962593, 988787

As for totients, there are 21 numbers in the range up to one million whose totient is a member of the Lucas sequence. These numbers are:

1, 2, 3, 4, 5, 6, 8, 10, 12, 19, 27, 38, 54, 2049, 2732, 4098, 5779, 11558, 36717, 48956, 73434

Another approach is to look at the determinant formed by the circulant matrix of a number. For example, 27255 has a circulant matrix as shown in Figure 1.

Figure 1

This matrix has a determinant of 21 which is a Fibonacci number. It turns out that there are 59500 numbers in the range up to one million that have this property (permalink).

Instead of the determinant, the permanent of the matrix could be considered. For example, 19140 has the circulant matrix shown in Figure 2.

Figure 2

In the range up to one million, there are only 68 such numbers (as opposed to the 59500 for the determinant). Here are the numbers:

[1, 2, 3, 5, 8, 10, 11, 12, 21, 22, 23, 32, 35, 53, 58, 85, 100, 101, 110, 200, 1000, 1001, 1021, 1100, 1102, 1120, 1201, 1222, 2011, 2022, 2110, 2122, 2202, 2212, 2220, 2221, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10419, 10941, 11000, 11001, 11010, 11094, 11100, 11490, 14019, 14901, 19104, 19140, 40191, 41109, 41910, 49011, 90114, 91041, 91401, 94110, 100000, 100001, 100100, 110000, 1000000]

Here are the Fibonacci numbers associated with each of these numbers:

1 --> 1

2 --> 2

3 --> 3

5 --> 5

8 --> 8

10 --> 1

11 --> 2

12 --> 5

21 --> 5

22 --> 8

23 --> 13

32 --> 13

35 --> 34

53 --> 34

58 --> 89

85 --> 89

100 --> 1

101 --> 2

110 --> 2

200 --> 8

1000 --> 1

1001 --> 2

1021 --> 34

1100 --> 2

1102 --> 34

1120 --> 34

1201 --> 34

1222 --> 233

2011 --> 34

2022 --> 144

2110 --> 34

2122 --> 233

2202 --> 144

2212 --> 233

2220 --> 144

2221 --> 233

10000 --> 1

10001 --> 2

10010 --> 2

10011 --> 13

10100 --> 2

10101 --> 13

10110 --> 13

10419 --> 75025

10941 --> 75025

11000 --> 2

11001 --> 13

11010 --> 13

11094 --> 75025

11100 --> 13

11490 --> 75025

14019 --> 75025

14901 --> 75025

19104 --> 75025

19140 --> 75025

40191 --> 75025

41109 --> 75025

41910 --> 75025

49011 --> 75025

90114 --> 75025

91041 --> 75025

91401 --> 75025

94110 --> 75025

100000 --> 1

100001 --> 2

100100 --> 8

110000 --> 2

The Circulant Matrix of a Number and its Permanent

In a post on December 15th 2022 titled Numbers and Matrices, I wrote about the circulant matrix of a number and looked at those special numbers where the determinant of the circulant matrix was equal to the number itself. An example of such a number is 494376160 which has the circulant matrix shown in Figure 1:

Figure 1

These sorts of numbers form OEIS A219324 and the initial members are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 247, 370, 378, 407, 481, 518, 592, 629, 1360, 3075, 26027, 26933, 45018, 69781, 80487, 154791, 1920261, 2137616, 2716713, 3100883, 3480140, 3934896, 4179451, 4830936, 5218958, 11955168, 80651025, 95738203, 257059332, 278945612, 456790123, 469135802, 493827160, 494376160

Instead of the determinant, we can use the permanent instead. The latter is not as well known as the former so Figure 2 provides a short explanation:

Figure 2

It's basically the same as the calculation for the determinant except that there are no alternating positive and negative signs. Figure 3 shows the calculations to determine the permanent of a rectangular matrix.

Figure 3

So now we can investigate what numbers have a circulant matrix whose permanent is equal to the number itself. It turns out that there are not that many. Here is the list up to ten million: 1, 2, 3, 4, 5, 6, 7, 8, 9, 261, 370, 407, 52036, 724212. The circulant matrix for 724212 is shown in Figure 4.

Figure 4

We can search for other numbers that the permanent might be equal to. The number 666 came to mind but if we collect the permanents of the first one million numbers, then there are only 269 numbers between 1 and 1000 (permalink). These are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 16, 17, 18, 20, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 41, 44, 45, 48, 49, 50, 52, 53, 54, 58, 61, 64, 65, 68, 72, 73, 74, 78, 79, 80, 81, 82, 84, 85, 89, 90, 91, 97, 98, 100, 106, 113, 116, 117, 120, 121, 125, 126, 128, 130, 132, 133, 136, 142, 144, 145, 152, 154, 155, 162, 164, 165, 168, 169, 171, 177, 178, 185, 189, 194, 198, 200, 201, 216, 217, 224, 226, 232, 233, 236, 243, 244, 250, 256, 257, 261, 265, 272, 273, 275, 280, 281, 289, 298, 299, 304, 305, 306, 314, 317, 324, 326, 337, 341, 343, 344, 349, 351, 353, 359, 361, 364, 366, 369, 370, 373, 384, 388, 389, 393, 394, 395, 396, 397, 400, 404, 407, 408, 409, 413, 416, 418, 420, 425, 432, 433, 434, 440, 443, 466, 468, 472, 481, 484, 486, 492, 493, 502, 504, 512, 513, 514, 520, 523, 526, 529, 530, 538, 539, 541, 544, 548, 559, 569, 574, 576, 577, 583, 586, 606, 612, 614, 624, 625, 626, 632, 633, 637, 638, 641, 656, 665, 673, 674, 676, 686, 691, 697, 706, 720, 728, 729, 730, 737, 745, 750, 753, 756, 757, 758, 765, 769, 772, 776, 782, 783, 784, 792, 793, 801, 807, 809, 810, 819, 824, 825, 833, 834, 838, 841, 850, 853, 854, 855, 872, 873, 877, 881, 882, 884, 885, 891, 896, 902, 907, 916, 926, 927, 928, 937, 945, 952, 953, 962, 964, 978, 980, 988, 990, 1000

As can be seen, 665 is among them but not 666. If we took the permanents of numbers in excess of one million, perhaps more numbers between 1 and 1000 would appear but it's doubtful. If we do the same for the determinants, we find that there are 417 of them between 1 and 1000. Oddly enough, 666 is not among them even though the numbers on either side (665 and 667) are (permalink).

Numbers and Matrices

Suppose we have a number with distinct, non-zero digits. The number associated with my diurnal age, 26918, is one such number. Let's now consider a 3 x 3 matrix with the numbers 1 to 9 positioned as shown in Figure 1:

Figure 1

Let's apply two rules to the digits in the matrix:

1. if a matrix digit does occur in the number then it is left unchanged 
2. if the matrix digit does not occur in the number then it is changed to a zero

See Figure 2 for the application of these two rules using 26918 as the number.

Figure 2: determinant is \(-\)48

Using this method, all permutations of the digits of a number will produce the same matrix e.g. 81962 (the reverse of 26918). Next find the determinant of the matrix. In the case of 26918, the determinant is \(-\)48. Does the determinant divide the number? Since 26918 factorises to 2 x 43 x 313, clearly it does not. 

However, there are numbers for which the determinant of its associated matrix does divide the number. Take for example 27468. It produces the matrix shown in Figure 3.

Figure 3: determinant is 84

This matrix has a determinant of 84 and this number appears in the divisors of 27468:

1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 109, 126, 218, 252, 327, 436, 654, 763, 981, 1308, 1526, 1962, 2289, 3052, 3924, 4578, 6867, 9156, 13734, 27468

How many numbers between 1 and 100,000 satisfy this divisibility rule? Well, not many. It turns out that there are only 256 numbers representing a mere 0.265% of the total range. The low number is not surprising as the number must firstly contain distinct non-zero digits and secondly the determinant of its associated matrix must divide the number. Here is the list of the numbers that pass the test:

384, 672, 735, 816, 1395, 1935, 1968, 3168, 3195, 3276, 3648, 3675, 3915, 4392, 6972, 9135, 9168, 9315, 9432, 12384, 12864, 13248, 13824, 13968, 14895, 14985, 15648, 17325, 17568, 17856, 18375, 18432, 18495, 18576, 18624, 18945, 19485, 19845, 21648, 21735, 21756, 24816, 27468, 28416, 28476, 29736, 31584, 31968, 34578, 34587, 34758, 34785, 34857, 34875, 35478, 35487, 35748, 35784, 35847, 35874, 36792, 37248, 37296, 37458, 37485, 37548, 37584, 37695, 37824, 37845, 37854, 38457, 38475, 38496, 38547, 38574, 38745, 38754, 39168, 39648, 41568, 41856, 41895, 41985, 42735, 42756, 42816, 43578, 43587, 43758, 43785, 43857, 43872, 43875, 43968, 45168, 45276, 45378, 45387, 45738, 45783, 45792, 45837, 45873, 46128, 46872, 47328, 47358, 47385, 47538, 47583, 47592, 47628, 47835, 47853, 47952, 48195, 48357, 48375, 48537, 48573, 48735, 48753, 48915, 49185, 49752, 49815, 51648, 53184, 53478, 53487, 53748, 53784, 53847, 53874, 54378, 54387, 54738, 54783, 54792, 54816, 54837, 54873, 57168, 57348, 57384, 57438, 57483, 57624, 57834, 57843, 58176, 58347, 58374, 58416, 58437, 58473, 58734, 58743, 59472, 61248, 61572, 61584, 61824, 62748, 64128, 65184, 67284, 67452, 67935, 71568, 71652, 71856, 72135, 72156, 72345, 72384, 73185, 73248, 73458, 73485, 73548, 73584, 73815, 73824, 73845, 73854, 74235, 74256, 74358, 74385, 74538, 74583, 74592, 74835, 74853, 74952, 75168, 75264, 75348, 75384, 75438, 75483, 75834, 75843, 76524, 78345, 78354, 78432, 78435, 78453, 78534, 78543, 78624, 79632, 81264, 81375, 81456, 81495, 81936, 81945, 82416, 83457, 83475, 83547, 83574, 83745, 83754, 84195, 84357, 84375, 84537, 84573, 84672, 84735, 84753, 84915, 85347, 85374, 85437, 85473, 85734, 85743, 87345, 87354, 87435, 87453, 87534, 87543, 89136, 89145, 89415, 91485, 91845, 92736, 93168, 93765, 94185, 94368, 94752, 94815, 95472, 96384, 98145, 98415

Here is the permalink to these results. This line of investigation was provoked by my inability to find much of interest about the number associated with my diurnal age, 26918. Most of what I found in the OEIS related to obscure properties connected with matrices. Ironically, I was then led to matrices anyway. Unfortunately, my number didn't qualify. 

To my delight however, I discovered that if I took the totient of 26918, which is 13104, then the determinant did divide it. The divisors of 13104 are listed below and it can be seen that 48 is a divisor:

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 18, 21, 24, 26, 28, 36, 39, 42, 48, 52, 56, 63, 72, 78, 84, 91, 104, 112, 117, 126, 144, 156, 168, 182, 208, 234, 252, 273, 312, 336, 364, 468, 504, 546, 624, 728, 819, 936, 1008, 1092, 1456, 1638, 1872, 2184, 3276, 4368, 6552, 13104

Our previous number, 27468, that was divisible by the determinant of its associated matrix,  has a totient of 7776 and this number is not divisible by the determinant. If we investigate how many numbers have their totients divisible by the determinant then, in the range up to 100,000, there are 2514 numbers that satisfy representing 2.514%. This is about ten times the number for divisibility of the number itself by the determinant. I won't list all the numbers but here is the permalink to the calculation. In the narrow range between 26900 and 27100, there are only two numbers that qualify: 26918 and 26957. 

While we are are considering totients, we may as well consider the sum of the divisors of the number. How many numbers have their sum of divisors divisible by their associated matrix's determinant. Well in the range, up to 100,000, there are 2864 such numbers, a little more than for the totients. Here is the permalink to the calculation. In the narrow range between 26900 and 27100, there are only five such numbers: 26937, 26945, 26974, 26975, 26978.

We can reduce these largish numbers (2514 for totients and 2864 for sum of divisors) by restricting our choice of numbers to primes. In that case, the totient will be one less than the prime number (and thus always composite) and the sum of divisors will be one more than the prime number (and thus always composite). For example, 23 has a totient of 22 and its sum of divisors is 24. 

Applying this restriction, we find that there are 66 prime numbers up to 100,000 that have their sum of divisors divisible by the determinant. This determinant must be non-zero and this is often not the case because even in five digit numbers there will be five zeroes in the matrix (and more in smaller numbers). In summary then, these numbers are not common because they must:

• be prime
• have no repeating digits
• contain no zeroes
• have a non-zero determinant
• have a determinant that divides their sum of digits

The numbers are:

1259, 2687, 8543, 13859, 14759, 14783, 15749, 18539, 23687, 25367, 26879, 28439, 29567, 31859, 35279, 37589, 41579, 41759, 42839, 45179, 48239, 49823, 51479, 51749, 51839, 51869, 53189, 53819, 56237, 56891, 57149, 57329, 58169, 58379, 58943, 61487, 61583, 65183, 65981, 68351, 68543, 71549, 74159, 75149, 75389, 78539, 78623, 81359, 81569, 81647, 82463, 83579, 84239, 85439, 85619, 85691, 86351, 86951, 87359, 89561, 89627, 92567, 94823, 96581, 96851, 98561

Let's take 1259 as an example. Its sum of divisors in 1260 which has the following divisors:

1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 28, 30, 35, 36, 42, 45, 60, 63, 70, 84, 90, 105, 126, 140, 180, 210, 252, 315, 420, 630, 1260

The matrix associated with this number has a determinant of 45 and this is one of the divisors. 

For the totients, there are only 50 prime numbers up to 100,000 that have their totients divisible by the determinant. Once again, in summary, these numbers are not common because they must:

• be prime
• have no repeating digits
• contain no zeroes
• have a non-zero determinant
• have a determinant that divides their totient

The numbers are:

3571, 4591, 4951, 6217, 6481, 7351, 7681, 8641, 13249, 21673, 24697, 25849, 26497, 26713, 31249, 31489, 34129, 39241, 39841, 42193, 42697, 45289, 47269, 47629, 48193, 48673, 49681, 52489, 59671, 62497, 63841, 65731, 67429, 72469, 72649, 73681, 76249, 79561, 81649, 82561, 82657, 83617, 83761, 84673, 84961, 93241, 94321, 97561, 97651, 98641

Let's take 98641 as an example. The associated matrix determinant is \(-\)48 and the totient is 98640 with divisors shown below (amongst which is the determinant):

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 137, 144, 180, 240, 274, 360, 411, 548, 685, 720, 822, 1096, 1233, 1370, 1644, 2055, 2192, 2466, 2740, 3288, 4110, 4932, 5480, 6165, 6576, 8220, 9864, 10960, 12330, 16440, 19728, 24660, 32880, 49320, 98640

I've added the three sequences (without the prime stipulation) to my Bespoken for Sequences on Google Docs (Figure 4, Figure 5 and Figure 6):

Figure 4: link

Figure 5: link

Figure 6: link

While these sequences may seem somewhat contrived they pale before some of the bizarre entries in the OEIS. Hence I'm emboldened to describe them here, although I have no intention of submitting them for approval by the venomous OEIS vetting committee. 

THE CIRCULANT MATRIX

If we want to get on a more mainstream connection between numbers and matrices, we need go no further than the circulant matrix which is a square matrix in which each row vector is rotated one element to the right relative to the preceding row vector. Thus a number like 26933 has this associated circulant matrix (see Figure 7):

Figure 7

What's interesting about this particular number is that its determinant is equal to the number itself. Numbers of this sort form OEIS A219324:

A219324

Positive integers n that are equal to the determinant of the circulant matrix formed by the decimal digits of n.

The initial members of the sequence are (permalink for numbers in the range up to 40,000):

1, 2, 3, 4, 5, 6, 7, 8, 9, 247, 370, 378, 407, 481, 518, 592, 629, 1360, 3075, 26027, 26933, 45018, 69781, 80487, 154791, 1920261, 2137616, 2716713, 3100883, 3480140, 3934896, 4179451, 4830936, 5218958, 11955168, 80651025, 95738203, 257059332, 278945612, 456790123, 469135802, 493827160, 494376160

I couldn't help broadening this criterion a little to accommodate the situation in which the determinant does not need to be equal to the number whose digits appear in the first row of the matrix but could be in any row. 25203 is such a number. Let's look at its circulant matrix (see Figure 8):

Figure 8

The determinant of this matrix is 3252 and this number, with a leading zero, appears in the second row. In the range up to 40,000, there are 69 numbers that satisfy including all 21 of the members of OEIS  A219324 in that range (marked in red and here is a permalink):

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 84, 100, 148, 185, 247, 259, 296, 370, 378, 407, 472, 481, 518, 592, 629, 703, 724, 740, 783, 814, 837, 851, 925, 962, 1000, 1360, 3075, 3150, 4002, 5031, 5071, 5471, 6013, 7150, 7154, 7530, 10000, 12054, 14063, 16072, 16978, 18450, 20325, 20541, 20596, 21607, 23035, 25203, 26027, 26933, 27260, 30352, 31406, 32520, 32693, 33269, 33604, 35230, 36043

Let's take one more example, this time 33269 with a circulant matrix as shown in Figure 9.

Figure 9

The determinant of this matrix is 26933 and looking at the matrix more closely we see that it is the same as the matrix shown in Figure 7 except that the top row has now moved to the bottom.  Thus it can be seen that, according to my new criterion, every row number is a member of the sequence in a circulant matrix where any one row number equals the derterminant. Thus 33269 is in the sequence and so too will be 69332 and 93326 (although I've only shown numbers up to 40,000).