A number whose digits are in strict descending order. more
The first 600 katadromes :
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
20,
21,
30,
31,
32,
40,
41,
42,
43,
50,
51,
52,
53,
54,
60,
61,
62,
63,
64,
65,
70,
71,
72,
73,
74,
75,
76,
80,
81,
82,
83,
84,
85,
86,
87,
90,
91,
92,
93,
94,
95,
96,
97,
98,
210,
310,
320,
321,
410,
420,
421,
430,
431,
432,
510,
520,
521,
530,
531,
532,
540,
541,
542,
543,
610,
620,
621,
630,
631,
632,
640,
641,
642,
643,
650,
651,
652,
653,
654,
710,
720,
721,
730,
731,
732,
740,
741,
742,
743,
750,
751,
752,
753,
754,
760,
761,
762,
763,
764,
765,
810,
820,
821,
830,
831,
832,
840,
841,
842,
843,
850,
851,
852,
853,
854,
860,
861,
862,
863,
864,
865,
870,
871,
872,
873,
874,
875,
876,
910,
920,
921,
930,
931,
932,
940,
941,
942,
943,
950,
951,
952,
953,
954,
960,
961,
962,
963,
964,
965,
970,
971,
972,
973,
974,
975,
976,
980,
981,
982,
983,
984,
985,
986,
987,
3210,
4210,
4310,
4320,
4321,
5210,
5310,
5320,
5321,
5410,
5420,
5421,
5430,
5431,
5432,
6210,
6310,
6320,
6321,
6410,
6420,
6421,
6430,
6431,
6432,
6510,
6520,
6521,
6530,
6531,
6532,
6540,
6541,
6542,
6543,
7210,
7310,
7320,
7321,
7410,
7420,
7421,
7430,
7431,
7432,
7510,
7520,
7521,
7530,
7531,
7532,
7540,
7541,
7542,
7543,
7610,
7620,
7621,
7630,
7631,
7632,
7640,
7641,
7642,
7643,
7650,
7651,
7652,
7653,
7654,
8210,
8310,
8320,
8321,
8410,
8420,
8421,
8430,
8431,
8432,
8510,
8520,
8521,
8530,
8531,
8532,
8540,
8541,
8542,
8543,
8610,
8620,
8621,
8630,
8631,
8632,
8640,
8641,
8642,
8643,
8650,
8651,
8652,
8653,
8654,
8710,
8720,
8721,
8730,
8731,
8732,
8740,
8741,
8742,
8743,
8750,
8751,
8752,
8753,
8754,
8760,
8761,
8762,
8763,
8764,
8765,
9210,
9310,
9320,
9321,
9410,
9420,
9421,
9430,
9431,
9432,
9510,
9520,
9521,
9530,
9531,
9532,
9540,
9541,
9542,
9543,
9610,
9620,
9621,
9630,
9631,
9632,
9640,
9641,
9642,
9643,
9650,
9651,
9652,
9653,
9654,
9710,
9720,
9721,
9730,
9731,
9732,
9740,
9741,
9742,
9743,
9750,
9751,
9752,
9753,
9754,
9760,
9761,
9762,
9763,
9764,
9765,
9810,
9820,
9821,
9830,
9831,
9832,
9840,
9841,
9842,
9843,
9850,
9851,
9852,
9853,
9854,
9860,
9861,
9862,
9863,
9864,
9865,
9870,
9871,
9872,
9873,
9874,
9875,
9876,
43210,
53210,
54210,
54310,
54320,
54321,
63210,
64210,
64310,
64320,
64321,
65210,
65310,
65320,
65321,
65410,
65420,
65421,
65430,
65431,
65432,
73210,
74210,
74310,
74320,
74321,
75210,
75310,
75320,
75321,
75410,
75420,
75421,
75430,
75431,
75432,
76210,
76310,
76320,
76321,
76410,
76420,
76421,
76430,
76431,
76432,
76510,
76520,
76521,
76530,
76531,
76532,
76540,
76541,
76542,
76543,
83210,
84210,
84310,
84320,
84321,
85210,
85310,
85320,
85321,
85410,
85420,
85421,
85430,
85431,
85432,
86210,
86310,
86320,
86321,
86410,
86420,
86421,
86430,
86431,
86432,
86510,
86520,
86521,
86530,
86531,
86532,
86540,
86541,
86542,
86543,
87210,
87310,
87320,
87321,
87410,
87420,
87421,
87430,
87431,
87432,
87510,
87520,
87521,
87530,
87531,
87532,
87540,
87541,
87542,
87543,
87610,
87620,
87621,
87630,
87631,
87632,
87640,
87641,
87642,
87643,
87650,
87651,
87652,
87653,
87654,
93210,
94210,
94310,
94320,
94321,
95210,
95310,
95320,
95321,
95410,
95420,
95421,
95430,
95431,
95432,
96210,
96310,
96320,
96321,
96410,
96420,
96421,
96430,
96431,
96432,
96510,
96520,
96521,
96530,
96531,
96532,
96540,
96541,
96542,
96543,
97210,
97310,
97320,
97321,
97410,
97420,
97421,
97430,
97431,
97432,
97510,
97520,
97521,
97530,
97531,
97532,
97540,
97541,
97542,
97543,
97610,
97620,
97621,
97630,
97631,
97632,
97640,
97641,
97642,
97643,
97650,
97651,
97652,
97653,
97654,
98210,
98310,
98320,
98321,
98410,
98420,
98421,
98430,
98431,
98432,
98510,
98520,
98521,
98530,
98531,
98532,
98540,
98541,
98542,
98543.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 1022 values, from 1 to 9876543210).
n\r | 0 | 1 |
2 | 681 | 341 | 2 |
3 | 350 | 336 | 336 | 3 |
4 | 272 | 205 | 409 | 136 | 4 |
5 | 527 | 264 | 132 | 66 | 33 | 5 |
6 | 231 | 114 | 228 | 119 | 222 | 108 | 6 |
7 | 142 | 145 | 149 | 148 | 146 | 146 | 146 | 7 |
8 | 160 | 117 | 233 | 56 | 112 | 88 | 176 | 80 | 8 |
9 | 118 | 112 | 112 | 116 | 112 | 112 | 116 | 112 | 112 | 9 |
10 | 511 | 256 | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | 10 |
11 | 0 | 9 | 37 | 93 | 162 | 210 | 210 | 162 | 93 | 37 | 9 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.