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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, 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
KEYWORD
nonn,tabf
AUTHOR
Paolo Xausa, Sep 13 2024
STATUS
approved