The alternating factorials up to 1015 :
1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, 1226280710981, 19696509177019, 335990918918981.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 10000000 values, from 1 to 1.2⋅1065657059).
| n\r | 0 | 1 | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 0 | 10000000 | 2 | ||||||||
| 3 | 0 | 5000001 | 4999999 | 3 | |||||||
| 4 | 0 | 5000001 | 0 | 4999999 | 4 | ||||||
| 5 | 1 | 5000000 | 0 | 0 | 4999999 | 5 | |||||
| 6 | 0 | 5000001 | 0 | 0 | 0 | 4999999 | 6 | ||||
| 7 | 0 | 2 | 0 | 4999999 | 4999997 | 2 | 0 | 7 | |||
| 8 | 0 | 2 | 0 | 4999999 | 0 | 4999999 | 0 | 0 | 8 | ||
| 9 | 0 | 3 | 4999998 | 0 | 0 | 1 | 0 | 4999998 | 0 | 9 | |
| 10 | 0 | 5000000 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 4999999 | 10 |
| 11 | 0 | 2 | 1 | 1 | 4999996 | 1 | 2 | 4999995 | 1 | 0 | 1 |
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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