A342767 Array T(n, k), n, k > 0, read by antidiagonals; a variant of lunar multiplication (A087062) based on prime factorizations of numbers (see Comments section for precise definition).

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 3, 4, 1, 1, 2, 4, 4, 2, 1, 1, 4, 3, 8, 3, 4, 1, 1, 2, 6, 4, 4, 6, 2, 1, 1, 8, 3, 8, 5, 8, 3, 8, 1, 1, 4, 8, 4, 6, 6, 4, 8, 4, 1, 1, 4, 9, 16, 5, 12, 5, 16, 9, 4, 1, 1, 2, 6, 8, 8, 6, 6, 8, 8, 6, 2, 1, 1, 8, 3, 8, 9, 16, 7, 16, 9, 8, 3, 8, 1

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,

- to "multiply" two prime numbers: take the smallest,

- to "add" two prime numbers: take the largest,

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

12 -> 2 2 3

14 -> x 2 7

-------

2 2 3

+ 2 2 2

---------

2 2 2 3 -> 24 = T(12, 14)

This sequence is closely related to lunar multiplication (A087062):

- 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 product in base p+1,

- as T(n, p) = n for any prime number >= A006530(n), we don't have prime numbers here,

- however, if we consider only p-smooth numbers (for some prime number p), then p is the "unit" and the semiprimes p*q (with q <= p) are "prime".

Examples

			Array T(n, k) begins:
  n\k|  1  2   3   4   5   6   7   8   9  10  11  12  13  14
  ---+------------------------------------------------------
    1|  1  1   1   1   1   1   1   1   1   1   1   1   1   1
    2|  1  2   2   4   2   4   2   8   4   4   2   8   2   4  ->  A061142
    3|  1  2   3   4   3   6   3   8   9   6   3  12   3   6  ->  A079065
    4|  1  4   4   8   4   8   4  16   8   8   4  16   4   8
    5|  1  2   3   4   5   6   5   8   9  10   5  12   5  10
    6|  1  4   6   8   6  12   6  16  18  12   6  24   6  12
    7|  1  2   3   4   5   6   7   8   9  10   7  12   7  14
    8|  1  8   8  16   8  16   8  32  16  16   8  32   8  16
    9|  1  4   9   8   9  18   9  16  27  18   9  36   9  18
   10|  1  4   6   8  10  12  10  16  18  20  10  24  10  20
   11|  1  2   3   4   5   6   7   8   9  10  11  12  11  14
   12|  1  8  12  16  12  24  12  32  36  24  12  48  12  24
   13|  1  2   3   4   5   6   7   8   9  10  11  12  13  14
   14|  1  4   6   8  10  12  14  16  18  20  14  24  14  28
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10011
• Rémy Sigrist, PARI program for A342767
• Index entries for sequences related to dismal (or lunar) arithmetic

Crossrefs

Cf. A006530, A061142, A079065, A087062, A342765, A342768.

Programs

PARI

See Links section.

Formula

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

T(n, n) = A342768(n).

T(n, 1) = 1.

T(n, 2) = A061142(n).

T(n, 3) = A079065(n).

T(n, p) = n for any prime number p >= A006530(n).