A number n such that d ⋅ nd contains a substring made of d digits d for a digit d = 2,...,9. more
The first 600 super-d numbers :
19,
31,
69,
81,
105,
106,
107,
119,
127,
131,
169,
181,
190,
219,
231,
247,
261,
269,
281,
310,
318,
319,
331,
332,
333,
334,
335,
336,
337,
338,
339,
348,
369,
381,
419,
431,
454,
462,
469,
471,
481,
511,
519,
531,
558,
569,
581,
601,
619,
631,
669,
679,
681,
690,
715,
719,
731,
739,
749,
753,
769,
781,
782,
783,
784,
810,
819,
831,
869,
881,
919,
928,
931,
944,
969,
981,
988,
1019,
1031,
1036,
1046,
1049,
1050,
1051,
1052,
1053,
1054,
1055,
1056,
1057,
1058,
1059,
1060,
1061,
1062,
1063,
1064,
1065,
1066,
1067,
1068,
1069,
1070,
1071,
1072,
1081,
1119,
1123,
1131,
1145,
1168,
1169,
1181,
1188,
1190,
1219,
1231,
1269,
1270,
1281,
1310,
1319,
1331,
1369,
1377,
1381,
1419,
1431,
1453,
1454,
1469,
1471,
1481,
1487,
1512,
1519,
1531,
1556,
1569,
1581,
1616,
1617,
1619,
1631,
1645,
1669,
1681,
1690,
1719,
1731,
1752,
1761,
1764,
1769,
1778,
1781,
1792,
1810,
1819,
1831,
1847,
1848,
1869,
1881,
1900,
1901,
1919,
1923,
1926,
1931,
1965,
1968,
1969,
1981,
1989,
2019,
2028,
2031,
2046,
2069,
2081,
2119,
2124,
2131,
2148,
2152,
2159,
2169,
2181,
2182,
2190,
2219,
2231,
2232,
2261,
2269,
2281,
2310,
2319,
2331,
2363,
2369,
2373,
2381,
2395,
2419,
2421,
2431,
2462,
2469,
2470,
2471,
2472,
2473,
2474,
2475,
2476,
2477,
2478,
2479,
2481,
2519,
2531,
2532,
2569,
2571,
2581,
2591,
2610,
2619,
2627,
2631,
2667,
2669,
2681,
2690,
2719,
2731,
2759,
2769,
2781,
2810,
2818,
2819,
2831,
2848,
2869,
2881,
2919,
2926,
2931,
2935,
2943,
2954,
2969,
2981,
3011,
3019,
3031,
3035,
3058,
3069,
3081,
3084,
3087,
3100,
3119,
3131,
3148,
3163,
3169,
3172,
3180,
3181,
3190,
3219,
3231,
3239,
3258,
3269,
3281,
3310,
3317,
3318,
3319,
3320,
3321,
3322,
3323,
3324,
3325,
3326,
3327,
3328,
3329,
3330,
3331,
3332,
3333,
3334,
3335,
3336,
3337,
3338,
3339,
3340,
3341,
3342,
3343,
3344,
3345,
3346,
3347,
3348,
3349,
3350,
3351,
3352,
3353,
3354,
3355,
3356,
3357,
3358,
3359,
3360,
3361,
3362,
3363,
3364,
3365,
3366,
3367,
3368,
3369,
3370,
3371,
3372,
3373,
3374,
3375,
3376,
3377,
3378,
3379,
3380,
3381,
3382,
3383,
3384,
3385,
3386,
3387,
3388,
3389,
3390,
3391,
3408,
3419,
3431,
3444,
3469,
3471,
3480,
3481,
3488,
3494,
3519,
3531,
3542,
3569,
3581,
3619,
3621,
3623,
3631,
3669,
3681,
3690,
3719,
3731,
3769,
3781,
3783,
3810,
3819,
3829,
3831,
3851,
3855,
3868,
3869,
3877,
3881,
3884,
3919,
3931,
3951,
3969,
3981,
4012,
4013,
4014,
4015,
4016,
4017,
4018,
4019,
4031,
4056,
4069,
4076,
4081,
4119,
4131,
4137,
4143,
4169,
4181,
4190,
4197,
4219,
4231,
4256,
4261,
4269,
4281,
4285,
4310,
4314,
4319,
4331,
4369,
4372,
4381,
4419,
4431,
4451,
4469,
4471,
4479,
4481,
4489,
4519,
4527,
4531,
4540,
4546,
4554,
4569,
4581,
4594,
4595,
4596,
4597,
4598,
4602,
4611,
4619,
4620,
4631,
4638,
4649,
4652,
4669,
4681,
4682,
4690,
4710,
4719,
4731,
4732,
4755,
4769,
4781,
4794,
4797,
4806,
4807,
4808,
4809,
4810,
4819,
4823,
4831,
4854,
4869,
4873,
4881,
4895,
4919,
4931,
4961,
4966,
4969,
4972,
4981,
5011,
5019,
5031,
5036,
5049,
5061,
5069,
5081,
5105,
5109,
5110,
5111,
5112,
5113,
5119,
5127,
5131,
5169,
5181,
5190,
5207,
5219,
5226,
5231,
5247,
5269,
5281,
5302,
5310,
5318,
5319,
5331,
5342,
5348,
5349,
5369,
5381,
5414,
5419,
5431,
5454,
5469,
5471,
5481,
5501,
5511,
5517,
5519,
5531,
5533,
5558,
5569,
5577,
5578,
5579,
5580,
5581,
5619,
5623,
5631,
5645,
5649,
5667,
5669,
5681,
5687,
5690,
5715,
5719,
5731,
5739,
5769,
5781,
5810,
5819,
5831,
5869,
5881,
5919,
5931,
5944,
5969,
5981,
5988,
6009,
6010,
6011,
6012,
6019,
6031,
6051,
6059,
6069,
6081,
6092,
6096,
6119,
6123,
6131,
6133,
6145,
6169,
6181.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 1516453 values, from 19 to 9999981).
n\r | 0 | 1 |
2 | 588269 | 928184 | 2 |
3 | 505703 | 505423 | 505327 | 3 |
4 | 278647 | 460983 | 309622 | 467201 | 4 |
5 | 242420 | 411717 | 228048 | 230582 | 403686 | 5 |
6 | 196005 | 309242 | 196083 | 309698 | 196181 | 309244 | 6 |
7 | 216899 | 216764 | 217163 | 216491 | 216288 | 216346 | 216502 | 7 |
8 | 137831 | 230203 | 154458 | 229223 | 140816 | 230780 | 155164 | 237978 | 8 |
9 | 169021 | 168360 | 168322 | 168246 | 168370 | 168477 | 168436 | 168693 | 168528 | 9 |
10 | 133409 | 300677 | 114897 | 113090 | 111431 | 109011 | 111040 | 113151 | 117492 | 292255 | 10 |
11 | 137686 | 137961 | 138167 | 137731 | 137734 | 137526 | 137726 | 138019 | 137731 | 138054 | 138118 |
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.