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.
LINKS
FORMULA
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 }
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Rémy Sigrist and N. J. A. Sloane, Apr 03 2021
STATUS
approved