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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).
5
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".
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
KEYWORD
nonn,tabl
AUTHOR
Rémy Sigrist, Apr 02 2021
STATUS
approved