|
|
A338797
|
|
Triangle read by rows: T(n,k) is the least m such that there exist positive integers x, y and z satisfying x/n + y/k = z/m where all fractions are reduced; 1 <= k <= n.
|
|
1
|
|
|
1, 2, 1, 3, 6, 1, 4, 4, 12, 1, 5, 10, 15, 20, 1, 6, 3, 2, 12, 30, 1, 7, 14, 21, 28, 35, 42, 1, 8, 8, 24, 8, 40, 24, 56, 1, 9, 18, 9, 36, 45, 18, 63, 72, 1, 10, 5, 30, 20, 2, 15, 70, 40, 90, 1, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
LINKS
|
|
|
FORMULA
|
T(n,k) = lcm(n,k) when gcd(n,k) = 1.
|
|
EXAMPLE
|
Table begins:
n\k| 1 2 3 4 5 6 7 8 9 10 11 12
---+-----------------------------------------------
1 | 1,
2 | 2, 1,
3 | 3, 6, 1,
4 | 4, 4, 12, 1,
5 | 5, 10, 15, 20, 1,
6 | 6, 3, 2, 12, 30, 1,
7 | 7, 14, 21, 28, 35, 42, 1,
8 | 8, 8, 24, 8, 40, 24, 56, 1,
9 | 9, 18, 9, 36, 45, 18, 63, 72, 1,
10 | 10, 5, 30, 20, 2, 15, 70, 40, 90, 1,
11 | 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 1,
12 | 12, 12, 4, 3, 60, 4, 84, 24, 36, 60, 132, 1.
T(20,10) = 4 because 1/20 + 7/10 = 3/4, and there is no choice of numerators on the left that results in a smaller denominator on the right.
|
|
PROG
|
(Haskell)
import Data.Ratio ((%), denominator)
farey n = [k % n | k <- [1..n], gcd n k == 1]
a338797T n k = minimum [denominator $ a + b | a <- farey n, b <- farey k]
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|