The rare numbers up to 1015 :
65, 621770, 281089082, 2022652202, 2042832002, 868591084757, 872546974178, 872568754178, 6979302951885, 20313693904202, 20313839704202, 20331657922202, 20331875722202, 20333875702202, 40313893704200, 40351893720200, 200142385731002, 204238494066002, 221462345754122, 244062891224042, 245518996076442, 248359494187442, 403058392434500, 441054594034340, 816984566129618.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 84 values, from 65 to 6.5⋅1019).
| n\r | 0 | 1 | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 66 | 18 | 2 | ||||||||
| 3 | 18 | 0 | 66 | 3 | |||||||
| 4 | 20 | 18 | 46 | 0 | 4 | ||||||
| 5 | 23 | 0 | 47 | 14 | 0 | 5 | |||||
| 6 | 15 | 0 | 51 | 3 | 0 | 15 | 6 | ||||
| 7 | 2 | 17 | 25 | 6 | 15 | 11 | 8 | 7 | |||
| 8 | 8 | 6 | 46 | 0 | 12 | 12 | 0 | 0 | 8 | ||
| 9 | 18 | 0 | 14 | 0 | 0 | 38 | 0 | 0 | 14 | 9 | |
| 10 | 11 | 0 | 41 | 0 | 0 | 12 | 0 | 6 | 14 | 0 | 10 |
| 11 | 41 | 0 | 5 | 0 | 0 | 0 | 7 | 16 | 4 | 0 | 11 |
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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