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

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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 7, 2, 1

Offset: 1

Author

Rémy Sigrist, Jun 03 2017

Keywords

• nonn
• tabl
• mult

Comments

We use the definition of recursive divisor given in A282446.

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).

This sequence has connections with the classical GCD (A003989).

For any i > 0, j > 0 and k > 0:

- T(i, j) = 1 iff gcd(i, j) = 1,

- A007947(T(i, j)) = A007947(gcd(i, j)),

- T(i, j) >= 1,

- T(i, j) <= min(i, j),

- T(i, j) <= gcd(i, j),

- T(i, 1) = 1,

- T(i, i) = i,

- T(i, j) = T(j, i) (the sequence is commutative),

- T(i, T(j, k)) = T(T(i, j), k) (the sequence is associative),

- T(i, i*j) <= i,

- if gcd(i, j) = 1 then T(i*j, k) = T(i, k) * T(j, k) (the sequence is multiplicative),

- T(i, 2*i) = A259445(i).

See also A287958 for the LCM equivalent.

Examples

			Table starts:
n\k|    1   2   3   4   5   6   7   8   9   10
---+-----------------------------------------------
1  |    1   1   1   1   1   1   1   1   1    1  ...
2  |    1   2   1   2   1   2   1   2   1    2  ...
3  |    1   1   3   1   1   3   1   1   3    1  ...
4  |    1   2   1   4   1   2   1   2   1    2  ...
5  |    1   1   1   1   5   1   1   1   1    5  ...
6  |    1   2   3   2   1   6   1   2   3    2  ...
7  |    1   1   1   1   1   1   7   1   1    1  ...
8  |    1   2   1   2   1   2   1   8   1    2  ...
9  |    1   1   3   1   1   3   1   1   9    1  ...
10 |    1   2   1   2   5   2   1   2   1   10  ...
...
T(4, 8) = T(2^2, 2^3) = 2.
		

Links

• Rémy Sigrist, First 100 antidiagonals of array, flattened
• Rémy Sigrist, Illustration of the first terms

Crossrefs

Cf. A003989, A007947, A182318, A259445, A282446, A287958.

Programs

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]))))