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

1,1

Links

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

Wikipedia, Optic equation.

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).

Example

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);
  ...

Mathematica

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

Crossrefs

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

Sequence in context: A317449 A222310 A294033 * A254827 A193862 A258631

Adjacent sequences: A376165 A376166 A376167 * A376169 A376170 A376171

Keyword

nonn,tabf

Author

Paolo Xausa, Sep 13 2024

Status

approved