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A216917 Square array read by antidiagonals, T(N,n) = lcm{1<=j<=N, gcd(j,n)=1 | j} for N >= 0, n >= 1. 4

%I

%S 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,

%T 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,

%U 2520,315,280,105,12,5,12,3,2,1,1,27720,315,280,105,84

%N Square array read by antidiagonals, T(N,n) = lcm{1<=j<=N, gcd(j,n)=1 | j} for N >= 0, n >= 1.

%C T(N,n) is the least common multiple of all integers up to N that are relatively prime to n.

%C Replacing LCM in the definition with "product" gives the Gauss factorial A216919.

%F For n > 0:

%F A(n,1) = A003418(n);

%F A(n,2^k) = A217858(n) for k > 0;

%F A(n,3^k) = A128501(n-1) for k > 0;

%F A(2,n) = A000034(n);

%F A(3,n) = A129203(n-1);

%F A(4,n) = A129197(n-1);

%F A(n,n) = A038610(n);

%F A(floor(n/2),n) = A124443(n);

%F A(n,1)/A(n,n) = A064446(n);

%F A(n,1)/A(n,2) = A053644(n).

%e n | N=0 1 2 3 4 5 6 7 8 9 10

%e -----+-------------------------------------

%e 1 | 1 1 2 6 12 60 60 420 840 2520 2520

%e 2 | 1 1 1 3 3 15 15 105 105 315 315

%e 3 | 1 1 2 2 4 20 20 140 280 280 280

%e 4 | 1 1 1 3 3 15 15 105 105 315 315

%e 5 | 1 1 2 6 12 12 12 84 168 504 504

%e 6 | 1 1 1 1 1 5 5 35 35 35 35

%e 7 | 1 1 2 6 12 60 60 60 120 360 360

%e 8 | 1 1 1 3 3 15 15 105 105 315 315

%e 9 | 1 1 2 2 4 20 20 140 280 280 280

%e 10 | 1 1 1 3 3 3 3 21 21 63 63

%e 11 | 1 1 2 6 12 60 60 420 840 2520 2520

%e 12 | 1 1 1 1 1 5 5 35 35 35 35

%e 13 | 1 1 2 6 12 60 60 420 840 2520 2520

%t 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-Fran├žois Alcover_, Jul 29 2013 *)

%o (Sage)

%o def A216917(N, n):

%o return lcm([j for j in (1..N) if gcd(j, n) == 1])

%o for n in (1..13): [A216917(N,n) for N in (0..10)]

%K nonn,tabl

%O 1,4

%A _Peter Luschny_, Oct 02 2012

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Last modified June 14 21:27 EDT 2021. Contains 345041 sequences. (Running on oeis4.)