A277227 Triangular array T read by rows: T(n,k) gives the additive orders k modulo n, for k = 0,1, ..., n-1.

1, 1, 2, 1, 3, 3, 1, 4, 2, 4, 1, 5, 5, 5, 5, 1, 6, 3, 2, 3, 6, 1, 7, 7, 7, 7, 7, 7, 1, 8, 4, 8, 2, 8, 4, 8, 1, 9, 9, 3, 9, 9, 3, 9, 9, 1, 10, 5, 10, 5, 2, 5, 10, 5, 10, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 12, 6, 4, 3, 12, 2, 12, 3, 4, 6, 12

Offset: 1

Author

Wolfdieter Lang, Oct 20 2016

Keywords

• nonn
• tabl
• easy

Comments

As a sequence A054531(n) = a(n+1), n >= 1.

As a triangular array this is the row reversed version of A054531.

The additive order of an element x of a group (G, +) is the least positive integer j with j*x := x + x + ... + x (j summands) = 0.

Equals A106448 when the first column (k = 0) of ones is removed. - Georg Fischer, Jul 26 2023

Examples

			The triangle begins:
n\k 0  1  2  3  4  5  6  7  8  9 10 11 ...
1:  1
2:  1  2
3:  1  3  3
4:  1  4  2  4
5:  1  5  5  5  5
6:  1  6  3  2  3  6
7:  1  7  7  7  7  7  7
8:  1  8  4  8  2  8  4  8
9:  1  9  9  3  9  9  3  9  9
10: 1 10  5 10  5  2  5 10  5 10
11: 1 11 11 11 11 11 11 11 11 11 11
12: 1 12  6  4  3 12  2 12  3  4  6 12
...
T(n, 0) = 1*0 = 0 = 0 (mod n), and n/GCD(n,0) = n/n = 1.
T(4, 2) = 2 because 2 + 2 = 4 = 0 (mod 4) and 2 is not 0 (mod 4).
T(4, 2) = n/GCD(2, 4) = 4/2 = 2.
		

Links

• Indranil Ghosh, Rows 1..100 of triangle, flattened

Crossrefs

Cf. A054531, A106448.

Formula

T(n, k) = order of the elements k of the finite abelian group (Z/(n Z), +), for k = 0, 1, ..., n-1.

T(n, k) = n/GCD(n, k), n >= 1, k = 0, 1, ..., n-1.

T(n, k) = A054531(n, n-k), n >=1, k = 0, 1, ..., n-1.