A287958 Table read by antidiagonals: T(n, k) = least recursive multiple of n and k; n > 0 and k > 0.

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, 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, 10, 9, 64, 35, 6, 35, 64, 9, 10, 11, 12, 22, 30, 36, 40, 42, 42, 40

Offset: 1

Author

Rémy Sigrist, Jun 03 2017

Keywords

• nonn
• tabl

Comments

We say that m is a recursive multiple of d iff d is a recursive divisor of m (as described in A282446).

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).

This sequence has connections with the classical LCM (A003990).

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

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

- T(i, j) >= 1,

- T(i, j) >= max(i, j),

- T(i, j) >= lcm(i, j),

- T(i, 1) = i,

- 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*j,

- if gcd(i, j) = 1 then T(i, j) = i*j.

See also A287957 for the GCD equivalent.

Examples

			Table starts:
n\k|     1   2   3   4   5   6   7   8   9  10
---+-----------------------------------------------
1  |     1   2   3   4   5   6   7   8   9  10  ...
2  |     2   2   6   4  10   6  14   8  18  10  ...
3  |     3   6   3  12  15   6  21  24   9  30  ...
4  |     4   4  12   4  20  12  28  64  36  20  ...
5  |     5  10  15  20   5  30  35  40  45  10  ...
6  |     6   6   6  12  30   6  42  24  18  30  ...
7  |     7  14  21  28  35  42   7  56  63  70  ...
8  |     8   8  24  64  40  24  56   8  72  40  ...
9  |     9  18   9  36  45  18  63  72   9  90  ...
10 |    10  10  30  20  10  30  70  40  90  10  ...
...
T(4, 8) = T(2^2, 2^3) = 2^(2*3) = 2^6 = 64.
		

Links

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

Crossrefs

Cf. A003990, A007947, A182318, A282446, A287957.

Programs

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]))))