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A277227
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Triangular array T read by rows: T(n,k) gives the additive orders k modulo n, for k = 0,1, ..., n-1.
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2
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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