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%I #12 Jun 04 2017 16:39:08
%S 1,2,2,3,2,3,4,6,6,4,5,4,3,4,5,6,10,12,12,10,6,7,6,15,4,15,6,7,8,14,6,
%T 20,20,6,14,8,9,8,21,12,5,12,21,8,9,10,18,24,28,30,30,28,24,18,10,11,
%U 10,9,64,35,6,35,64,9,10,11,12,22,30,36,40,42,42,40
%N Table read by antidiagonals: T(n, k) = least recursive multiple of n and k; n > 0 and k > 0.
%C We say that m is a recursive multiple of d iff d is a recursive divisor of m (as described in A282446).
%C More informally, the prime tower factorization of T(n, k) is the union of the prime tower factorizations of n and k (the prime tower factorization of a number is defined in A182318).
%C This sequence has connections with the classical LCM (A003990).
%C For any i > 0, j > 0 and k > 0:
%C - A007947(T(i, j)) = A007947(lcm(i, j)),
%C - T(i, j) >= 1,
%C - T(i, j) >= max(i, j),
%C - T(i, j) >= lcm(i, j),
%C - T(i, 1) = i,
%C - T(i, i) = i,
%C - T(i, j) = T(j, i) (the sequence is commutative),
%C - T(i, T(j, k)) = T(T(i, j), k) (the sequence is associative),
%C - T(i, i*j) >= i*j,
%C - if gcd(i, j) = 1 then T(i, j) = i*j.
%C See also A287957 for the GCD equivalent.
%H Rémy Sigrist, <a href="/A287958/b287958.txt">First 100 antidiagonals of array, flattened</a>
%H Rémy Sigrist, <a href="/A287958/a287958.pdf">Illustration of the first terms</a>
%e Table starts:
%e n\k| 1 2 3 4 5 6 7 8 9 10
%e ---+-----------------------------------------------
%e 1 | 1 2 3 4 5 6 7 8 9 10 ...
%e 2 | 2 2 6 4 10 6 14 8 18 10 ...
%e 3 | 3 6 3 12 15 6 21 24 9 30 ...
%e 4 | 4 4 12 4 20 12 28 64 36 20 ...
%e 5 | 5 10 15 20 5 30 35 40 45 10 ...
%e 6 | 6 6 6 12 30 6 42 24 18 30 ...
%e 7 | 7 14 21 28 35 42 7 56 63 70 ...
%e 8 | 8 8 24 64 40 24 56 8 72 40 ...
%e 9 | 9 18 9 36 45 18 63 72 9 90 ...
%e 10 | 10 10 30 20 10 30 70 40 90 10 ...
%e ...
%e T(4, 8) = T(2^2, 2^3) = 2^(2*3) = 2^6 = 64.
%o (PARI) T(n,k) = if (n*k==0, return (max(n,k))); my (g=factor(lcm(n,k))); return (prod(i=1, #g~, g[i,1]^T(valuation(n, g[i,1]), valuation(k, g[i,1]))))
%Y Cf. A003990, A007947, A182318, A282446, A287957.
%K nonn,tabl
%O 1,2
%A _Rémy Sigrist_, Jun 03 2017
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