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

1,5

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".

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

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).

Example

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

Prog

(PARI) See Links section.

Crossrefs

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

Sequence in context: A101566 A331447 A352460 * A176653 A174842 A156074

Adjacent sequences: A342764 A342765 A342766 * A342768 A342769 A342770

Keyword

nonn,tabl

Author

Rémy Sigrist, Apr 02 2021

Status

approved