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A287957 Table read by antidiagonals: T(n, k) = greatest common recursive divisor of n and k; n > 0 and k > 0. 3

%I

%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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Last modified June 20 09:30 EDT 2021. Contains 345162 sequences. (Running on oeis4.)