A number n such that n2 ends with the digits of n. more
The automorphic numbers up to 10
15 :
1,
5,
6,
25,
76,
376,
625,
9376,
90625,
109376,
890625,
2890625,
7109376,
12890625,
87109376,
212890625,
787109376,
1787109376,
8212890625,
18212890625,
81787109376,
918212890625,
9918212890625,
40081787109376,
59918212890625,
259918212890625,
740081787109376.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 27 values, from 1 to 740081787109376).
n\r | 0 | 1 |
2 | 12 | 15 | 2 |
3 | 7 | 11 | 9 | 3 |
4 | 11 | 15 | 1 | 0 | 4 |
5 | 14 | 13 | 0 | 0 | 0 | 5 |
6 | 4 | 6 | 3 | 3 | 5 | 6 | 6 |
7 | 1 | 4 | 4 | 5 | 4 | 5 | 4 | 7 |
8 | 10 | 14 | 0 | 0 | 1 | 1 | 1 | 0 | 8 |
9 | 0 | 1 | 0 | 3 | 5 | 4 | 4 | 5 | 5 | 9 |
10 | 0 | 1 | 0 | 0 | 0 | 14 | 12 | 0 | 0 | 0 | 10 |
11 | 1 | 4 | 3 | 3 | 1 | 3 | 3 | 1 | 3 | 2 | 3 |
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.