The straight-line numbers up to 1015 :
111, 123, 135, 147, 159, 210, 222, 234, 246, 258, 321, 333, 345, 357, 369, 420, 432, 444, 456, 468, 531, 543, 555, 567, 579, 630, 642, 654, 666, 678, 741, 753, 765, 777, 789, 840, 852, 864, 876, 888, 951, 963, 975, 987, 999, 1111, 1234, 1357, 2222, 2345, 2468, 3210, 3333, 3456, 3579, 4321, 4444, 4567, 5432, 5555, 5678, 6420, 6543, 6666, 6789, 7531, 7654, 7777, 8642, 8765, 8888, 9630, 9753, 9876, 9999, 11111, 12345, 13579, 22222, 23456, 33333, 34567, 43210, 44444, 45678, 54321, 55555, 56789, 65432, 66666, 76543, 77777, 86420, 87654, 88888, 97531, 98765, 99999, 111111, 123456, 222222, 234567, 333333, 345678, 444444, 456789, 543210, 555555, 654321, 666666, 765432, 777777, 876543, 888888, 987654, 999999, 1111111, 1234567, 2222222, 2345678, 3333333, 3456789, 4444444, 5555555, 6543210, 6666666, 7654321, 7777777, 8765432, 8888888, 9876543, 9999999, 11111111, 12345678, 22222222, 23456789, 33333333, 44444444, 55555555, 66666666, 76543210, 77777777, 87654321, 88888888, 98765432, 99999999, 111111111, 123456789, 222222222, 333333333, 444444444, 555555555, 666666666, 777777777, 876543210, 888888888, 987654321, 999999999, 1111111111, 2222222222, 3333333333, 4444444444, 5555555555, 6666666666, 7777777777, 8888888888, 9876543210, 9999999999, 11111111111, 22222222222, 33333333333, 44444444444, 55555555555, 66666666666, 77777777777, 88888888888, 99999999999, 111111111111, 222222222222, 333333333333, 444444444444, 555555555555, 666666666666, 777777777777, 888888888888, 999999999999, 1111111111111, 2222222222222, 3333333333333, 4444444444444, 5555555555555, 6666666666666, 7777777777777, 8888888888888, 9999999999999, 11111111111111, 22222222222222, 33333333333333, 44444444444444, 55555555555555, 66666666666666, 77777777777777, 88888888888888, 99999999999999, 111111111111111, 222222222222222, 333333333333333, 444444444444444, 555555555555555, 666666666666666, 777777777777777, 888888888888888, 999999999999999.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 213 values, from 111 to 999999999999999).
| n\r | 0 | 1 | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 98 | 115 | 2 | ||||||||
| 3 | 136 | 38 | 39 | 3 | |||||||
| 4 | 46 | 52 | 52 | 63 | 4 | ||||||
| 5 | 35 | 45 | 44 | 44 | 45 | 5 | |||||
| 6 | 60 | 22 | 22 | 76 | 16 | 17 | 6 | ||||
| 7 | 42 | 24 | 28 | 28 | 29 | 31 | 31 | 7 | |||
| 8 | 25 | 26 | 26 | 23 | 21 | 26 | 26 | 40 | 8 | ||
| 9 | 51 | 14 | 13 | 42 | 10 | 10 | 43 | 14 | 16 | 9 | |
| 10 | 14 | 25 | 22 | 22 | 20 | 21 | 20 | 22 | 22 | 25 | 10 |
| 11 | 54 | 8 | 17 | 19 | 25 | 24 | 18 | 20 | 14 | 14 | 0 |
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.
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