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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.

%I A376168 #23 Sep 15 2024 02:14:49
%S A376168 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,
%T A376168 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,
%U A376168 72,10,40,12,24,16,16,24,12,40,10,72,9,10,90,12,36,18,18,36,12,90,10
%N 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.
%H A376168 Paolo Xausa, <a href="/A376168/b376168.txt">Table of n, a(n) for n = 1..14340</a> (rows 1..400 of triangle, flattened).
%H A376168 Wikipedia, <a href="https://en.wikipedia.org/wiki/Optic_equation">Optic equation</a>.
%F A376168 T(n,1) = T(n,2*A048691(n)) = n + 1.
%F A376168 T(n,A048691(n)) = T(n,A048691(n) + 1) = n*2.
%F A376168 T(n,k) = T(n,2*A048691(n) - k + 1), with 1 <= k <= 2*A048691(n).
%e A376168 Triangle begins:
%e A376168   [1] ( 2, 2);
%e A376168   [2] ( 3, 6),( 4, 4),( 6, 3);
%e A376168   [3] ( 4,12),( 6, 6),(12, 4);
%e A376168   [4] ( 5,20),( 6,12),( 8, 8),(12, 6),(20, 5);
%e A376168   [5] ( 6,30),(10,10),(30, 6);
%e A376168   [6] ( 7,42),( 8,24),( 9,18),(10,15),(12,12),(15,10),(18,9),(24,8),(42,7);
%e A376168   [7] ( 8,56),(14,14),(56, 8);
%e A376168   [8] ( 9,72),(10,40),(12,24),(16,16),(24,12),(40,10),(72,9);
%e A376168   [9] (10,90),(12,36),(18,18),(36,12),(90,10);
%e A376168   ...
%t A376168 A376168row[n_] := Module[{a, b}, SolveValues[1/a + 1/b == 1/n && a > 0 && b > 0, {a, b}, Integers]];
%t A376168 Array[A376168row, 10]
%Y A376168 Cf. A018892, A048691 (row lengths/2), A376169 (row sums).
%K A376168 nonn,tabf
%O A376168 1,1
%A A376168 _Paolo Xausa_, Sep 13 2024