VOOZH about

URL: https://oeis.org/A343033

⇱ A343033 - OEIS

login
A343033
Array T(n, k), n, k > 0, read by antidiagonals; a variant of lunar multiplication (A087062) based on prime exponents of numbers (see Comments section for precise definition).
2
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 5, 2, 1, 1, 5, 3, 3, 5, 1, 1, 6, 7, 4, 7, 6, 1, 1, 7, 15, 5, 5, 15, 7, 1, 1, 2, 11, 6, 11, 6, 11, 2, 1, 1, 3, 3, 7, 35, 35, 7, 3, 3, 1, 1, 10, 5, 4, 13, 30, 13, 4, 5, 10, 1, 1, 11, 21, 9, 5, 77, 77, 5, 9, 21, 11, 1
OFFSET
1,5
COMMENTS
To compute T(n, k):
- write the prime exponents of n and of k on two lines, right aligned (these lines correspond to rows of A067255 in reversed order),
- to "multiply" two prime numbers: take the smallest,
- to "add" two prime numbers: take the largest,
- for example, for T(12, 14):
(11 7 5 3 2)
12 --> 1 2
14 --> x 1 0 0 1
---------
1 1
0 0
0 0
+ 1 1
-----------
1 1 0 1 1 --> 462 = T(12, 14)
This sequence is closely related to lunar multiplication (A087062):
- for any b > 1, let S_b be the set of nonnegative integers m such that A051903(m)< b,
- there is a natural bijection f from S_b to the set of nonnegative integers:
f(Product_{k >= 0} prime(k)^d(k)) = Sum_{k >= 0} d(k) * b^k,
- let g be the inverse of f,
- then for any numbers n and k in S_b, we have:
T(n, k) = g(f(n) "*" f(k)) where "*" denotes lunar product in base b,
- the corresponding addition table is A003990.
FORMULA
T(n, k) = T(k, n).
T(n, 1) = 1.
T(n, 2) = A007947(n).
T(n, 3) = A328915(n).
T(n, 4) = A007948(n).
T(n, n) = A343035(n).
A051903(T(n, k)) = min(A051903(n), A051903(k)).
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 3 2 5 6 7 2 3 10 11 6 13 14 --> A007947
3| 1 3 5 3 7 15 11 3 5 21 13 15 17 33 --> A328915
4| 1 2 3 4 5 6 7 4 9 10 11 12 13 14 --> A007948
5| 1 5 7 5 11 35 13 5 7 55 17 35 19 65
6| 1 6 15 6 35 30 77 6 15 210 143 30 221 462
7| 1 7 11 7 13 77 17 7 11 91 19 77 23 119
8| 1 2 3 4 5 6 7 8 9 10 11 12 13 14
9| 1 3 5 9 7 15 11 9 25 21 13 45 17 33
10| 1 10 21 10 55 210 91 10 21 110 187 210 247 910
11| 1 11 13 11 17 143 19 11 13 187 23 143 29 209
12| 1 6 15 12 35 30 77 12 45 210 143 60 221 462
13| 1 13 17 13 19 221 23 13 17 247 29 221 31 299
14| 1 14 33 14 65 462 119 14 33 910 209 462 299 238
PROG
(PARI) T(n, k) = { my (r=1, pp=factor(n)[, 1]~, qq=factor(k)[, 1]~); for (i=1, #pp, for (j=1, #qq, my (p=prime(primepi(pp[i])+primepi(qq[j])-1), v=valuation(r, p), w=min(valuation(n, pp[i]), valuation(k, qq[j]))); if (w>v, r*=p^(w-v)))); r }
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved