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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.

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