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A216917
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Square array read by antidiagonals, T(N,n) = lcm{1<=j<=N, gcd(j,n)=1 | j} for N >= 0, n >= 1.
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4
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1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 12, 3, 2, 1, 1, 60, 3, 2, 1, 1, 1, 60, 15, 4, 3, 2, 1, 1, 420, 15, 20, 3, 6, 1, 1, 1, 840, 105, 20, 15, 12, 1, 2, 1, 1, 2520, 105, 140, 15, 12, 1, 6, 1, 1, 1, 2520, 315, 280, 105, 12, 5, 12, 3, 2, 1, 1, 27720, 315, 280, 105, 84
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OFFSET
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1,4
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COMMENTS
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T(N,n) is the least common multiple of all integers up to N that are relatively prime to n.
Replacing LCM in the definition with "product" gives the Gauss factorial A216919.
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LINKS
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FORMULA
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For n > 0:
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EXAMPLE
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n | N=0 1 2 3 4 5 6 7 8 9 10
-----+-------------------------------------
1 | 1 1 2 6 12 60 60 420 840 2520 2520
2 | 1 1 1 3 3 15 15 105 105 315 315
3 | 1 1 2 2 4 20 20 140 280 280 280
4 | 1 1 1 3 3 15 15 105 105 315 315
5 | 1 1 2 6 12 12 12 84 168 504 504
6 | 1 1 1 1 1 5 5 35 35 35 35
7 | 1 1 2 6 12 60 60 60 120 360 360
8 | 1 1 1 3 3 15 15 105 105 315 315
9 | 1 1 2 2 4 20 20 140 280 280 280
10 | 1 1 1 3 3 3 3 21 21 63 63
11 | 1 1 2 6 12 60 60 420 840 2520 2520
12 | 1 1 1 1 1 5 5 35 35 35 35
13 | 1 1 2 6 12 60 60 420 840 2520 2520
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MATHEMATICA
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t[_, 0] = 1; t[n_, k_] := LCM @@ Select[Range[k], CoprimeQ[#, n]&]; Table[t[n - k + 1, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)
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PROG
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(Sage)
return lcm([j for j in (1..N) if gcd(j, n) == 1])
for n in (1..13): [A216917(N, n) for N in (0..10)]
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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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