A316586 Array read by antidiagonals: T(n,k) is the number of elements x in SL(2,Z_n) with x^k == I mod n where I is the identity matrix.
1, 1, 1, 1, 4, 1, 1, 3, 2, 1, 1, 4, 9, 8, 1, 1, 1, 8, 9, 2, 1, 1, 6, 1, 32, 21, 8, 1, 1, 1, 18, 1, 32, 27, 2, 1, 1, 4, 1, 24, 25, 32, 57, 16, 1, 1, 3, 8, 1, 42, 1, 44, 33, 2, 1, 1, 4, 9, 32, 1, 108, 1, 160, 99, 8, 1, 1, 1, 2, 9, 32, 1, 114, 1, 56, 63, 2, 1
Offset: 1
Examples
Array begins:
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n\k | 1 2 3 4 5 6 7 8 9 10
------+-----------------------------------------
1 | 1 1 1 1 1 1 1 1 1 1 ...
2 | 1 4 3 4 1 6 1 4 3 4 ...
3 | 1 2 9 8 1 18 1 8 9 2 ...
4 | 1 8 9 32 1 24 1 32 9 8 ...
5 | 1 2 21 32 25 42 1 32 21 50 ...
6 | 1 8 27 32 1 108 1 32 27 8 ...
7 | 1 2 57 44 1 114 49 128 57 2 ...
8 | 1 16 33 160 1 144 1 256 33 16 ...
9 | 1 2 99 56 1 198 1 56 243 2 ...
10 | 1 8 63 128 25 252 1 128 63 200 ...
11 | 1 2 111 112 265 222 1 112 111 530 ...
12 | 1 16 81 256 1 432 1 256 81 16 ...
13 | 1 2 183 184 1 366 469 184 183 2 ...
14 | 1 8 171 176 1 684 49 512 171 8 ...
15 | 1 4 189 256 25 756 1 256 189 100 ...
...
Formula
T(n,k) = Sum_{d|k} A316564(n, d).
Conjecture: T(p,p) = p^2 for p prime.
Comments