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%I #18 Nov 21 2020 00:59:35
%S 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,
%T 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,
%U 11,22,33,44,55,66,77,88,99,110,1
%N 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.
%H Peter Kagey, <a href="/A338797/b338797.txt">Table of n, a(n) for n = 1..10011</a> (first 141 rows, flattened)
%F A051537(n,k) <= T(n,k) <= A221918(n,k) <= lcm(n,k) = A051173(n,k).
%F T(n,k) = lcm(n,k) when gcd(n,k) = 1.
%e Table begins:
%e n\k| 1 2 3 4 5 6 7 8 9 10 11 12
%e ---+-----------------------------------------------
%e 1 | 1,
%e 2 | 2, 1,
%e 3 | 3, 6, 1,
%e 4 | 4, 4, 12, 1,
%e 5 | 5, 10, 15, 20, 1,
%e 6 | 6, 3, 2, 12, 30, 1,
%e 7 | 7, 14, 21, 28, 35, 42, 1,
%e 8 | 8, 8, 24, 8, 40, 24, 56, 1,
%e 9 | 9, 18, 9, 36, 45, 18, 63, 72, 1,
%e 10 | 10, 5, 30, 20, 2, 15, 70, 40, 90, 1,
%e 11 | 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 1,
%e 12 | 12, 12, 4, 3, 60, 4, 84, 24, 36, 60, 132, 1.
%e 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.
%o (Haskell)
%o import Data.Ratio ((%), denominator)
%o farey n = [k % n | k <- [1..n], gcd n k == 1]
%o a338797T n k = minimum [denominator $ a + b | a <- farey n, b <- farey k]
%Y Cf. A051173, A051537, A221918.
%K nonn,tabl
%O 1,2
%A _Peter Kagey_, Nov 09 2020