A342765 Array T(n, k), n, k > 0, read by antidiagonals; T(n, k) = max(A006530(n), A006530(k)) * T(n/A006530(n), k/A006530(k)) with T(1, 1) = 1.

1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 5, 4, 3, 4, 5, 6, 5, 6, 6, 5, 6, 7, 6, 5, 4, 5, 6, 7, 8, 7, 6, 10, 10, 6, 7, 8, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 9, 12, 14, 10, 10, 14, 12, 9, 10, 11, 10, 9, 8, 7, 6, 7, 8, 9, 10, 11, 12, 11, 10, 9, 20, 14, 14, 20, 9, 10, 11, 12

Offset: 1

Author

Rémy Sigrist, Apr 02 2021

Keywords

• nonn
• tabl

Comments

To compute T(n, k):

- write the prime factors of n and of k in ascending order with multiplicities on two lines, right aligned,

- take the largest prime number in each column and multiply back,

- for example, for T(12, 14):

12 -> 2 2 3

14 -> 2 7

-----

2 2 7 -> 28 = T(12, 14)

This sequence is closely related to lunar addition (A087061):

- let n and k be two p-smooth numbers,

- let f be the function that associates to a p-smooth number, say m, the unique number whose (p+1)-base digits are prime, nondecreasing and whose product is m,

- let g be the inverse of f,

- then for any p-smooth numbers n and k, T(n, k) = g(f(n) "+" f(k)) where "+" denotes lunar addition in base p+1,

- see A342767 for the corresponding multiplication.

Examples

			Array T(n, k) begins:
  n\k|   1   2   3   4   5   6   7   8   9  10  11  12  13  14
  ---+--------------------------------------------------------
    1|   1   2   3   4   5   6   7   8   9  10  11  12  13  14
    2|   2   2   3   4   5   6   7   8   9  10  11  12  13  14
    3|   3   3   3   6   5   6   7  12   9  10  11  12  13  14
    4|   4   4   6   4  10   6  14   8   9  10  22  12  26  14
    5|   5   5   5  10   5  10   7  20  15  10  11  20  13  14
    6|   6   6   6   6  10   6  14  12   9  10  22  12  26  14
    7|   7   7   7  14   7  14   7  28  21  14  11  28  13  14
    8|   8   8  12   8  20  12  28   8  18  20  44  12  52  28
    9|   9   9   9   9  15   9  21  18   9  15  33  18  39  21
   10|  10  10  10  10  10  10  14  20  15  10  22  20  26  14
   11|  11  11  11  22  11  22  11  44  33  22  11  44  13  22
   12|  12  12  12  12  20  12  28  12  18  20  44  12  52  28
   13|  13  13  13  26  13  26  13  52  39  26  13  52  13  26
   14|  14  14  14  14  14  14  14  28  21  14  22  28  26  14
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10011
• Index entries for sequences related to dismal (or lunar) arithmetic

Crossrefs

Cf. A001222, A006530, A087061, A342766, A342767.

Programs

PARI

gpf(n) = if (n==1, 1, my (p=factor(n)[,1]~); p[#p])
T(n, k) = if (n==1 || k==1, max(n, k), my (p=gpf(n), q=gpf(k)); max(p, q)*T(n/p, k/q))

Formula

T(n, k) = T(k, n).

T(m, T(n, k)) = T(T(m, n), k).

T(n, 1) = n.

T(n, n) = n.

A001222(T(n, k)) = max(A001222(n), A001222(k)).

A006530(T(n, k)) = max(A006530(n), A006530(k)).