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A250211
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Square array read by antidiagonals: A(m,n) = multiplicative order of m mod n, or 0 if m and n are not coprime.
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3
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1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 0, 0, 1, 1, 1, 1, 2, 4, 1, 1, 0, 2, 0, 4, 0, 1, 1, 1, 0, 1, 2, 0, 3, 1, 1, 0, 1, 0, 0, 0, 6, 0, 1, 1, 1, 2, 2, 1, 2, 3, 2, 6, 1, 1, 0, 0, 0, 4, 0, 6, 0, 0, 0, 1, 1, 1, 1, 1, 4, 1, 2, 2, 3, 4, 10, 1, 1, 0, 2, 0, 2, 0, 0, 0, 6, 0, 5, 0, 1, 1, 1, 0, 2, 0, 0, 1, 2, 0, 0, 5, 0, 12, 1
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OFFSET
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1,9
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COMMENTS
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Read by antidiagonals:
m\n 1 2 3 4 5 6 7 8 9 10 11 12 13
1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 0 2 0 4 0 3 0 6 0 10 0 12
3 1 1 0 2 4 0 6 2 0 4 5 0 3
4 1 0 1 0 2 0 3 0 3 0 5 0 6
5 1 1 2 1 0 2 6 2 6 0 5 2 4
6 1 0 0 0 1 0 2 0 0 0 10 0 12
7 1 1 1 2 4 1 0 2 3 4 10 2 12
8 1 0 2 0 4 0 1 0 2 0 10 0 4
9 1 1 0 1 2 0 3 1 0 2 5 0 3
10 1 0 1 0 0 0 6 0 1 0 2 0 6
11 1 1 2 2 1 2 3 2 6 1 0 2 12
12 1 0 0 0 4 0 6 0 0 0 1 0 2
13 1 1 1 1 4 1 2 2 3 4 10 1 0
etc.
A(m,n) = Least k>0 such that m^k=1 (mod n), or 0 if no such k exists.
It is easy to prove that column n has period n.
A(1,n) = 1, A(m,1) =1.
If A(m,n) differs from 0, it is period length of 1/n in base m.
The maximum number in column n is psi(n) (A002322(n)), and all numbers in column n (except 0) divide psi(n), and all factors of psi(n) are in column n.
Except the first row, every row contains all natural numbers.
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LINKS
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EXAMPLE
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A(3,7) = 6 because:
3^0 = 1 (mod 7)
3^1 = 3 (mod 7)
3^2 = 2 (mod 7)
3^3 = 6 (mod 7)
3^4 = 4 (mod 7)
3^5 = 5 (mod 7)
3^6 = 1 (mod 7)
...
And the period is 6, so A(3,7) = 6.
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MAPLE
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f:= proc(m, n)
if igcd(m, n) <> 1 then 0
elif n=1 then 1
else numtheory:-order(m, n)
fi
end proc:
seq(seq(f(t-j, j), j=1..t-1), t=2..65); # Robert Israel, Dec 30 2014
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MATHEMATICA
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a250211[m_, n_] = If[GCD[m, n] == 1, MultiplicativeOrder[m, n], 0]
Table[a250211[t-j, j], {t, 2, 65}, {j, 1, t-1}]
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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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