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

1,5

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.

Links

Rémy Sigrist, First 100 antidiagonals of array, flattened

Rémy Sigrist, Illustration of the first terms

Example

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.

Prog

(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]))))

Crossrefs

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

Sequence in context: A353282 A350714 A159923 * A003989 A091255 A332013

Adjacent sequences: A287954 A287955 A287956 * A287958 A287959 A287960

Keyword

nonn,tabl,mult

Author

Rémy Sigrist, Jun 03 2017

Status

approved