A376168 Irregular triangle read by rows: row n lists all of the integer pairs (a,b) such that 1/a + 1/b = 1/n, sorted by a.

2, 2, 3, 6, 4, 4, 6, 3, 4, 12, 6, 6, 12, 4, 5, 20, 6, 12, 8, 8, 12, 6, 20, 5, 6, 30, 10, 10, 30, 6, 7, 42, 8, 24, 9, 18, 10, 15, 12, 12, 15, 10, 18, 9, 24, 8, 42, 7, 8, 56, 14, 14, 56, 8, 9, 72, 10, 40, 12, 24, 16, 16, 24, 12, 40, 10, 72, 9, 10, 90, 12, 36, 18, 18, 36, 12, 90, 10

Offset: 1

Author

Paolo Xausa, Sep 13 2024

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  [1] ( 2, 2);
  [2] ( 3, 6),( 4, 4),( 6, 3);
  [3] ( 4,12),( 6, 6),(12, 4);
  [4] ( 5,20),( 6,12),( 8, 8),(12, 6),(20, 5);
  [5] ( 6,30),(10,10),(30, 6);
  [6] ( 7,42),( 8,24),( 9,18),(10,15),(12,12),(15,10),(18,9),(24,8),(42,7);
  [7] ( 8,56),(14,14),(56, 8);
  [8] ( 9,72),(10,40),(12,24),(16,16),(24,12),(40,10),(72,9);
  [9] (10,90),(12,36),(18,18),(36,12),(90,10);
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..14340 (rows 1..400 of triangle, flattened).
• Wikipedia, Optic equation.

Crossrefs

Cf. A018892, A048691 (row lengths/2), A376169 (row sums).

Programs

Mathematica

A376168row[n_] := Module[{a, b}, SolveValues[1/a + 1/b == 1/n && a > 0 && b > 0, {a, b}, Integers]];
Array[A376168row, 10]

Formula

T(n,1) = T(n,2*A048691(n)) = n + 1.

T(n,A048691(n)) = T(n,A048691(n) + 1) = n*2.

T(n,k) = T(n,2*A048691(n) - k + 1), with 1 <= k <= 2*A048691(n).