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, 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

Author

Peter Kagey, Nov 09 2020

Keywords

• nonn
• tabl

Examples

			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.
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)

Crossrefs

Cf. A051173, A051537, A221918.

Programs

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]

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.