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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
OFFSET
1,2
LINKS
Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)
FORMULA
A051537(n,k) <= T(n,k) <= A221918(n,k) <= lcm(n,k) = A051173(n,k).
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
nonn,tabl
AUTHOR
Peter Kagey, Nov 09 2020
STATUS
approved