The impolite numbers up to 1015 :
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472, 274877906944, 549755813888, 1099511627776, 2199023255552, 4398046511104, 8796093022208, 17592186044416, 35184372088832, 70368744177664, 140737488355328, 281474976710656, 562949953421312.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 50 values, from 1 to 562949953421312).
| n\r | 0 | 1 | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 49 | 1 | 2 | ||||||||
| 3 | 0 | 25 | 25 | 3 | |||||||
| 4 | 48 | 1 | 1 | 0 | 4 | ||||||
| 5 | 0 | 13 | 13 | 12 | 12 | 5 | |||||
| 6 | 0 | 1 | 25 | 0 | 24 | 0 | 6 | ||||
| 7 | 0 | 17 | 17 | 0 | 16 | 0 | 0 | 7 | |||
| 8 | 47 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 8 | ||
| 9 | 0 | 9 | 9 | 0 | 8 | 8 | 0 | 8 | 8 | 9 | |
| 10 | 0 | 1 | 13 | 0 | 12 | 0 | 12 | 0 | 12 | 0 | 10 |
| 11 | 0 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 |
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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