A number whose digits go down and then up, with only one digit in the valley. more
The first 600 canyon numbers :
101,
102,
103,
104,
105,
106,
107,
108,
109,
201,
202,
203,
204,
205,
206,
207,
208,
209,
212,
213,
214,
215,
216,
217,
218,
219,
301,
302,
303,
304,
305,
306,
307,
308,
309,
312,
313,
314,
315,
316,
317,
318,
319,
323,
324,
325,
326,
327,
328,
329,
401,
402,
403,
404,
405,
406,
407,
408,
409,
412,
413,
414,
415,
416,
417,
418,
419,
423,
424,
425,
426,
427,
428,
429,
434,
435,
436,
437,
438,
439,
501,
502,
503,
504,
505,
506,
507,
508,
509,
512,
513,
514,
515,
516,
517,
518,
519,
523,
524,
525,
526,
527,
528,
529,
534,
535,
536,
537,
538,
539,
545,
546,
547,
548,
549,
601,
602,
603,
604,
605,
606,
607,
608,
609,
612,
613,
614,
615,
616,
617,
618,
619,
623,
624,
625,
626,
627,
628,
629,
634,
635,
636,
637,
638,
639,
645,
646,
647,
648,
649,
656,
657,
658,
659,
701,
702,
703,
704,
705,
706,
707,
708,
709,
712,
713,
714,
715,
716,
717,
718,
719,
723,
724,
725,
726,
727,
728,
729,
734,
735,
736,
737,
738,
739,
745,
746,
747,
748,
749,
756,
757,
758,
759,
767,
768,
769,
801,
802,
803,
804,
805,
806,
807,
808,
809,
812,
813,
814,
815,
816,
817,
818,
819,
823,
824,
825,
826,
827,
828,
829,
834,
835,
836,
837,
838,
839,
845,
846,
847,
848,
849,
856,
857,
858,
859,
867,
868,
869,
878,
879,
901,
902,
903,
904,
905,
906,
907,
908,
909,
912,
913,
914,
915,
916,
917,
918,
919,
923,
924,
925,
926,
927,
928,
929,
934,
935,
936,
937,
938,
939,
945,
946,
947,
948,
949,
956,
957,
958,
959,
967,
968,
969,
978,
979,
989,
1012,
1013,
1014,
1015,
1016,
1017,
1018,
1019,
1023,
1024,
1025,
1026,
1027,
1028,
1029,
1034,
1035,
1036,
1037,
1038,
1039,
1045,
1046,
1047,
1048,
1049,
1056,
1057,
1058,
1059,
1067,
1068,
1069,
1078,
1079,
1089,
2012,
2013,
2014,
2015,
2016,
2017,
2018,
2019,
2023,
2024,
2025,
2026,
2027,
2028,
2029,
2034,
2035,
2036,
2037,
2038,
2039,
2045,
2046,
2047,
2048,
2049,
2056,
2057,
2058,
2059,
2067,
2068,
2069,
2078,
2079,
2089,
2101,
2102,
2103,
2104,
2105,
2106,
2107,
2108,
2109,
2123,
2124,
2125,
2126,
2127,
2128,
2129,
2134,
2135,
2136,
2137,
2138,
2139,
2145,
2146,
2147,
2148,
2149,
2156,
2157,
2158,
2159,
2167,
2168,
2169,
2178,
2179,
2189,
3012,
3013,
3014,
3015,
3016,
3017,
3018,
3019,
3023,
3024,
3025,
3026,
3027,
3028,
3029,
3034,
3035,
3036,
3037,
3038,
3039,
3045,
3046,
3047,
3048,
3049,
3056,
3057,
3058,
3059,
3067,
3068,
3069,
3078,
3079,
3089,
3101,
3102,
3103,
3104,
3105,
3106,
3107,
3108,
3109,
3123,
3124,
3125,
3126,
3127,
3128,
3129,
3134,
3135,
3136,
3137,
3138,
3139,
3145,
3146,
3147,
3148,
3149,
3156,
3157,
3158,
3159,
3167,
3168,
3169,
3178,
3179,
3189,
3201,
3202,
3203,
3204,
3205,
3206,
3207,
3208,
3209,
3212,
3213,
3214,
3215,
3216,
3217,
3218,
3219,
3234,
3235,
3236,
3237,
3238,
3239,
3245,
3246,
3247,
3248,
3249,
3256,
3257,
3258,
3259,
3267,
3268,
3269,
3278,
3279,
3289,
4012,
4013,
4014,
4015,
4016,
4017,
4018,
4019,
4023,
4024,
4025,
4026,
4027,
4028,
4029,
4034,
4035,
4036,
4037,
4038,
4039,
4045,
4046,
4047,
4048,
4049,
4056,
4057,
4058,
4059,
4067,
4068,
4069,
4078,
4079,
4089,
4101,
4102,
4103,
4104,
4105,
4106,
4107,
4108,
4109,
4123,
4124,
4125,
4126,
4127,
4128,
4129,
4134,
4135,
4136,
4137,
4138,
4139,
4145,
4146,
4147,
4148,
4149,
4156,
4157,
4158,
4159,
4167,
4168,
4169,
4178,
4179,
4189,
4201,
4202,
4203,
4204,
4205,
4206,
4207,
4208,
4209,
4212,
4213,
4214,
4215,
4216,
4217,
4218,
4219,
4234,
4235,
4236,
4237,
4238.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 346484 values, from 101 to 987654323456789).
n\r | 0 | 1 |
2 | 115465 | 231019 | 2 |
3 | 115555 | 115424 | 115505 | 3 |
4 | 46226 | 138607 | 69239 | 92412 | 4 |
5 | 10881 | 22287 | 44822 | 89668 | 178826 | 5 |
6 | 38508 | 76984 | 38517 | 77047 | 38440 | 76988 | 6 |
7 | 49502 | 49497 | 49492 | 49498 | 49499 | 49495 | 49501 | 7 |
8 | 27170 | 59918 | 29878 | 38101 | 19056 | 78689 | 39361 | 54311 | 8 |
9 | 38519 | 38470 | 38506 | 38516 | 38474 | 38498 | 38520 | 38480 | 38501 | 9 |
10 | 0 | 511 | 1277 | 2681 | 5425 | 10881 | 21776 | 43545 | 86987 | 173401 | 10 |
11 | 51894 | 40940 | 27430 | 15755 | 9107 | 9059 | 15638 | 27375 | 41040 | 52039 | 56207 |
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.