I noticed that the number associated with my diurnal age today has a circulant matrix with a determinant that is a pronic number. The number in question is 27514. This is its circulant matrix. Forgive the formatting as I just copied it from the SageMathCell output.
[2 7 5 2 4]
[4 2 7 5 2]
[2 4 2 7 5]
[5 2 4 2 7]
[7 5 2 4 2]
This matrix has a determinant of 10100 = 100 x 101 and thus pronic. Naturally I wondered how many other numbers between 1 and 40000 share this property and it turns out that 502 do so that's about 1.25% of the range. Here are the numbers (permalink):
2, 6, 20, 42, 60, 64, 93, 95, 101, 113, 131, 134, 143, 200, 204, 279, 297, 309, 311, 314, 341, 347, 374, 377, 399, 402, 413, 420, 431, 437, 446, 464, 473, 479, 497, 600, 640, 644, 677, 729, 734, 737, 743, 749, 767, 773, 776, 789, 792, 794, 798, 879, 897, 903, 927, 930, 939, 947, 950, 972, 974, 978, 987, 993, 1010, 1041, 1130, 1310, 1340, 1430, 2000, 2040, 2253, 2352, 2790, 2970, 3090, 3110, 3140, 3144, 3151, 3296, 3410, 3441, 3470, 3692, 3740, 3770, 3990, 4002, 4011, 4020, 4042, 4130, 4134, 4187, 4192, 4200, 4291, 4310, 4370, 4431, 4460, 4640, 4730, 4781, 4790, 4970, 5061, 5131, 5223, 5322, 6000, 6051, 6374, 6385, 6400, 6440, 6473, 6583, 6770, 7290, 7340, 7364, 7370, 7430, 7463, 7490, 7670, 7730, 7760, 7797, 7890, 7920, 7940, 7980, 8147, 8365, 8389, 8563, 8688, 8697, 8741, 8790, 8796, 8886, 8970, 8983, 9003, 9007, 9030, 9142, 9236, 9241, 9270, 9296, 9300, 9390, 9399, 9470, 9500, 9632, 9687, 9692, 9720, 9740, 9777, 9780, 9786, 9799, 9870, 9930, 9993, 9997, 10001, 10100, 10122, 10227, 10254, 10317, 10397, 10410, 10694, 10731, 10795, 11073, 11112, 11121, 11202, 11211, 11300, 11343, 12021, 12072, 12111, 12333, 12342, 12405, 12432, 12557, 12698, 12702, 13017, 13100, 13224, 13233, 13314, 13323, 13332, 13367, 13400, 13431, 13486, 13553, 13583, 13646, 13701, 13709, 13736, 13853, 14133, 14223, 14300, 14366, 14638, 15006, 15042, 15275, 15335, 15338, 15725, 16005, 16373, 16409, 16463, 16634, 16667, 16676, 16766, 16829, 16843, 17103, 17509, 17552, 17633, 17666, 18335, 18364, 18962, 19046, 19057, 19073, 19286, 20000, 20004, 20145, 20172, 20211, 20400, 20495, 20721, 21012, 21111, 21207, 21234, 21243, 21333, 21504, 21755, 21896, 22017, 22101, 22233, 22314, 22323, 22332, 22358, 22378, 22385, 22413, 22495, 22530, 22547, 22583, 22594, 22745, 22853, 22873, 23133, 23142, 23223, 23232, 23313, 23322, 23331, 23421, 23458, 23520, 23528, 23548, 23582, 23782, 23825, 23827, 23852, 24051, 24132, 24257, 24275, 24321, 24385, 24509, 24529, 24758, 24835, 24952, 24987, 25157, 25238, 25283, 25328, 25384, 25429, 25472, 25487, 25559, 25571, 25595, 25724, 25832, 25834, 25942, 25955, 26778, 26981, 27021, 27102, 27238, 27283, 27452, 27515, 27524, 27687, 27845, 27867, 27894, 27900, 28235, 28253, 28325, 28327, 28453, 28479, 28532, 28543, 28574, 28619, 28732, 28776, 29054, 29168, 29245, 29254, 29555, 29700, 29748, 30171, 30179, 30566, 30575, 30665, 30900, 31100, 31107, 31134, 31233, 31323, 31332, 31355, 31358, 31385, 31400, 31413, 31422, 31440, 31510, 31637, 31646, 31684, 31763, 31907, 32124, 32133, 32223, 32232, 32241, 32258, 32285, 32287, 32313, 32322, 32331, 32528, 32728, 32825, 32845, 32854, 32960, 33123, 33132, 33141, 33176, 33213, 33222, 33231, 33312, 33321, 33335, 33353, 33515, 33518, 33533, 33569, 33671, 33789, 33815, 33965, 33987, 34100, 34166, 34212, 34258, 34311, 34410, 34582, 34599, 34700, 34861, 35153, 35183, 35248, 35282, 35333, 35482, 35507, 35531, 35606, 35693, 35822, 35831, 35936, 35949, 35996, 35999, 36056, 36065, 36137, 36359, 36395, 36418, 36461, 36506, 36614, 36713, 36920, 36995, 37011, 37055, 37091, 37316, 37361, 37400, 37700, 37799, 37822, 37893, 37938, 37979, 38146, 38153, 38252, 38272, 38379, 38397, 38425, 38522, 38524, 38531, 39495, 39536, 39569, 39599, 39653, 39659, 39738, 39797, 39873, 39900, 39954, 39959, 39977, 39995
The determinants can be equal to anything we please, for example prime numbers. Up to 40000, there are 196 numbers with circulant matrices having determinants that are prime numbers. All these primes lie between 2 and 29. The numbers are (permalink):
2, 3, 5, 7, 20, 21, 30, 32, 43, 50, 65, 70, 76, 98, 101, 122, 200, 210, 212, 221, 223, 232, 300, 320, 322, 344, 430, 434, 443, 445, 454, 500, 544, 566, 650, 656, 665, 667, 676, 700, 760, 766, 788, 878, 887, 980, 1010, 1011, 1121, 1220, 2000, 2100, 2111, 2120, 2122, 2210, 2221, 2230, 2320, 3000, 3200, 3220, 3233, 3332, 3343, 3440, 4300, 4333, 4340, 4430, 4450, 4454, 4540, 5000, 5440, 5444, 5455, 5554, 5660, 6500, 6560, 6566, 6650, 6665, 6670, 6760, 7000, 7600, 7660, 7787, 7880, 8777, 8780, 8788, 8870, 8887, 9800, 10001, 10011, 10100, 10101, 10110, 11001, 11122, 11210, 11212, 11221, 11696, 12112, 12121, 12200, 12211, 12332, 12442, 13223, 13663, 14224, 16336, 16619, 16961, 19166, 20000, 20023, 20302, 21000, 21110, 21112, 21121, 21200, 21211, 21220, 21233, 21244, 22100, 22111, 22210, 22223, 22232, 22300, 22313, 22322, 22333, 22414, 22454, 23132, 23200, 23222, 23233, 23321, 23323, 23332, 24142, 24421, 24425, 24542, 25244, 30000, 30035, 30503, 31322, 31366, 32000, 32002, 32123, 32200, 32222, 32231, 32233, 32323, 32330, 32332, 33212, 33223, 33232, 33320, 33322, 33344, 33430, 33434, 33443, 33454, 33616, 34334, 34343, 34400, 34433, 34435, 34444, 34543, 34664, 35344, 35885, 36163, 36446, 36631, 38558
An example is 25244 with the following circulant matrix having a determinant of 17:
[2 5 2 4 4]
[4 2 5 2 4]
[4 4 2 5 2]
[2 4 4 2 5]
[5 2 4 4 2]
What about numbers with circulant matrices that have determinants with digits that are permutations of the number's digits? There are 130 numbers that satisfy this criterion. They are (permalink):
1, 2, 3, 4, 5, 6, 7, 8, 9, 84, 148, 158, 184, 185, 247, 259, 269, 274, 295, 296, 307, 378, 387, 407, 418, 427, 472, 481, 518, 529, 581, 592, 629, 692, 703, 704, 724, 738, 742, 783, 814, 815, 837, 841, 851, 873, 925, 926, 952, 962, 1063, 3075, 5174, 5471, 6013, 7035, 7154, 7451, 10548, 12348, 13824, 14085, 14283, 14669, 15804, 16496, 16946, 16978, 17689, 18432, 18796, 19664, 19867, 20627, 20749, 20762, 21843, 22076, 22639, 22936, 23269, 23296, 23396, 23481, 23639, 23778, 23987, 24097, 24138, 26027, 26392, 26702, 26923, 26933, 27206, 27387, 27837, 27893, 27904, 28314, 28379, 28773, 29362, 29363, 29623, 29738, 31428, 32184, 32692, 32693, 32789, 32877, 32936, 32962, 33269, 33962, 34812, 36229, 36329, 36392, 37278, 37782, 37928, 38241, 38297, 38727, 39226, 39236, 39623, 39872
An example is 27837 with a determinant of 23787. Here is a full list of the numbers and their determinants. Determinants that are equal to their associated numbers are highlighted (omitting the trivial single digit numbers - permalink):
1 --> 1
2 --> 2
3 --> 3
4 --> 4
5 --> 5
6 --> 6
7 --> 7
8 --> 8
9 --> 9
84 --> 48
148 --> 481
158 --> 518
184 --> 481
185 --> 518
247 --> 247
259 --> 592
269 --> 629
274 --> 247
295 --> 592
296 --> 629
307 --> 370
378 --> 378
387 --> 378
407 --> 407
418 --> 481
427 --> 247
472 --> 247
481 --> 481
518 --> 518
529 --> 592
581 --> 518
592 --> 592
629 --> 629
692 --> 629
703 --> 370
704 --> 407
724 --> 247
738 --> 378
742 --> 247
783 --> 378
814 --> 481
815 --> 518
837 --> 378
841 --> 481
851 --> 518
873 --> 378
925 --> 592
926 --> 629
952 --> 592
962 --> 629
1063 --> 1360
3075 --> 3075
5174 --> 1547
5471 --> 1547
6013 --> 1360
7035 --> 3075
7154 --> 1547
7451 --> 1547
10548 --> 45018
12348 --> 23418
13824 --> 23418
14085 --> 45018
14283 --> 23418
14669 --> 49166
15804 --> 45018
16496 --> 49166
16946 --> 49166
16978 --> 69781
17689 --> 69781
18432 --> 23418
18796 --> 69781
19664 --> 49166
19867 --> 69781
20627 --> 26027
20749 --> 97042
20762 --> 26027
21843 --> 23418
22076 --> 26027
22639 --> 39622
22936 --> 39622
23269 --> 39622
23296 --> 39622
23396 --> 26933
23481 --> 23418
23639 --> 26933
23778 --> 23787
23987 --> 28739
24097 --> 97042
24138 --> 23418
26027 --> 26027
26392 --> 39622
26702 --> 26027
26923 --> 39622
26933 --> 26933
27206 --> 26027
27387 --> 23787
27837 --> 23787
27893 --> 28739
27904 --> 97042
28314 --> 23418
28379 --> 28739
28773 --> 23787
29362 --> 39622
29363 --> 26933
29623 --> 39622
29738 --> 28739
31428 --> 23418
32184 --> 23418
32692 --> 39622
32693 --> 26933
32789 --> 28739
32877 --> 23787
32936 --> 26933
32962 --> 39622
33269 --> 26933
33962 --> 26933
34812 --> 23418
36229 --> 39622
36329 --> 26933
36392 --> 26933
37278 --> 23787
37782 --> 23787
37928 --> 28739
38241 --> 23418
38297 --> 28739
38727 --> 23787
39226 --> 39622
39236 --> 26933
39623 --> 26933
39872 --> 28739
In summary then there are 19 numbers with determinants equal to the number itself, with 10 being non-trivial. These are (permalink):
1, 2, 3, 4, 5, 6, 7, 8, 9, 247, 378, 407, 481, 518, 592, 629, 3075, 26027, 26933
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