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%I #14 Mar 08 2020 00:07:12
%S 1,3,3,4,7,4,7,12,12,7,6,15,10,15,6,12,18,28,28,18,12,8,28,24,27,24,
%T 28,8,15,24,30,42,42,30,24,15,13,31,32,60,16,60,32,31,13,18,39,60,56,
%U 72,72,56,60,39,18,12,42,28,51,48,70,48,51,28,42,12,28,36
%N Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} lcm(c,d) for n>=1, k>=1.
%C T(n,n) = A064950(n) = sum_{d|n} d*tau(d^2).
%C T(n,1) = T(1,n) = A000203(n) = sigma(n).
%C T(n,2) = T(2,n) = A062731(n) = sigma(2*n).
%C T(n+1,n) = A083539(n) = sigma(n+1)*sigma(n).
%C T(prime(n),1) = A008864(n) = prime(n)+1.
%H Alois P. Heinz, <a href="/A216626/b216626.txt">Antidiagonals n = 1..141, flattened</a>
%e [-----1---2---3----4----5----6----7----8----9---10---11---12]
%e [ 1] 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28
%e [ 2] 3, 7, 12, 15, 18, 28, 24, 31, 39, 42, 36, 60
%e [ 3] 4, 12, 10, 28, 24, 30, 32, 60, 28, 72, 48, 70
%e [ 4] 7, 15, 28, 27, 42, 60, 56, 51, 91, 90, 84, 108
%e [ 5] 6, 18, 24, 42, 16, 72, 48, 90, 78, 48, 72, 168
%e [ 6] 12, 28, 30, 60, 72, 70, 96, 124, 84, 168, 144, 150
%e [ 7] 8, 24, 32, 56, 48, 96, 22, 120, 104, 144, 96, 224
%e [ 8] 15, 31, 60, 51, 90, 124, 120, 83, 195, 186, 180, 204
%e [ 9] 13, 39, 28, 91, 78, 84, 104, 195, 55, 234, 156, 196
%e [10] 18, 42, 72, 90, 48, 168, 144, 186, 234, 112, 216, 360
%e [11] 12, 36, 48, 84, 72, 144, 96, 180, 156, 216, 34, 336
%e [12] 28, 60, 70, 108, 168, 150, 224, 204, 196, 360, 336, 270
%e .
%e Displayed as a triangular array:
%e 1;
%e 3, 3;
%e 4, 7, 4;
%e 7, 12, 12, 7;
%e 6, 15, 10, 15, 6;
%e 12, 18, 28, 28, 18, 12;
%e 8, 28, 24, 27, 24, 28, 8;
%e 15, 24, 30, 42, 42, 30, 24, 15;
%e 13, 31, 32, 60, 16, 60, 32, 31, 13;
%p with(numtheory):
%p T:= (n, k) -> add(add(ilcm(c, d), c=divisors(n)), d=divisors(k)):
%p seq (seq (T(n, 1+d-n), n=1..d), d=1..12); # _Alois P. Heinz_, Sep 12 2012
%t T[n_, k_] := Sum[LCM[c, d], {c, Divisors[n]}, {d, Divisors[k]}]; Table[T[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Mar 25 2014 *)
%o (Sage)
%o def A216626(n, k) :
%o cp = cartesian_product([divisors(n), divisors(k)])
%o return reduce(lambda x,y: x+y, map(lcm, cp))
%o for n in (1..12): [A216626(n,k) for k in (1..12)]
%Y Cf. A216620, A216621, A216622, A216623, A216624, A216625, A216627.
%K nonn,tabl
%O 1,2
%A _Peter Luschny_, Sep 12 2012
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