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%I #14 Jun 04 2017 16:38:44
%S 1,1,1,1,2,1,1,1,1,1,1,2,3,2,1,1,1,1,1,1,1,1,2,1,4,1,2,1,1,1,3,1,1,3,
%T 1,1,1,2,1,2,5,2,1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,3,2,1,6,1,2,3,2,1,1,1,
%U 1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,2,7,2,1
%N Table read by antidiagonals: T(n, k) = greatest common recursive divisor of n and k; n > 0 and k > 0.
%C We use the definition of recursive divisor given in A282446.
%C More informally, the prime tower factorization of T(n, k) is the intersection 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 GCD (A003989).
%C For any i > 0, j > 0 and k > 0:
%C - T(i, j) = 1 iff gcd(i, j) = 1,
%C - A007947(T(i, j)) = A007947(gcd(i, j)),
%C - T(i, j) >= 1,
%C - T(i, j) <= min(i, j),
%C - T(i, j) <= gcd(i, j),
%C - T(i, 1) = 1,
%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,
%C - if gcd(i, j) = 1 then T(i*j, k) = T(i, k) * T(j, k) (the sequence is multiplicative),
%C - T(i, 2*i) = A259445(i).
%C See also A287958 for the LCM equivalent.
%H Rémy Sigrist, <a href="/A287957/b287957.txt">First 100 antidiagonals of array, flattened</a>
%H Rémy Sigrist, <a href="/A287957/a287957.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 1 1 1 1 1 1 1 1 1 ...
%e 2 | 1 2 1 2 1 2 1 2 1 2 ...
%e 3 | 1 1 3 1 1 3 1 1 3 1 ...
%e 4 | 1 2 1 4 1 2 1 2 1 2 ...
%e 5 | 1 1 1 1 5 1 1 1 1 5 ...
%e 6 | 1 2 3 2 1 6 1 2 3 2 ...
%e 7 | 1 1 1 1 1 1 7 1 1 1 ...
%e 8 | 1 2 1 2 1 2 1 8 1 2 ...
%e 9 | 1 1 3 1 1 3 1 1 9 1 ...
%e 10 | 1 2 1 2 5 2 1 2 1 10 ...
%e ...
%e T(4, 8) = T(2^2, 2^3) = 2.
%o (PARI) T(n,k) = my (g=factor(gcd(n,k))); return (prod(i=1, #g~, g[i,1]^T(valuation(n, g[i,1]), valuation(k, g[i,1]))))
%Y Cf. A003989, A007947, A182318, A259445, A282446, A287958.
%K nonn,tabl,mult
%O 1,5
%A _Rémy Sigrist_, Jun 03 2017
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