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%I #7 Sep 20 2021 07:10:20
%S 1,3,1,7,1,1,11,3,1,1,19,3,1,1,1,23,7,3,1,1,1,35,7,3,1,1,1,1,43,11,3,
%T 3,1,1,1,1,55,11,7,3,1,1,1,1,1,63,19,7,3,3,1,1,1,1,1,83,19,7,3,3,1,1,
%U 1,1,1,1,91,23,11,7,3,3,1,1,1,1,1,1,115,23
%N Triangle read by rows: T(n,k) = number of pairs (x,y) in {1..n}X{1..n} with gcd(x,y) = k.
%C T(n,1) = A018805(n);
%C sum(T(n,k): k = 1..n) = A000290(n);
%C sum(T(n,k): k = 2..n) = A100613(n);
%C T(floor(n/k),1) = A018805(n).
%H Reinhard Zumkeller, <a href="/A242114/b242114.txt">Rows n = 1..125 of table, flattened</a>
%F T(n,k) = A018805(A010766(n,k));
%e T(4,1) = #{(1,1), (1,2), (1,3), (1,4), (2,1), (2,3), (3,1), (3,2), (3,4), (4,1), (4,3)} = 11;
%e T(4,2) = #{(2,2), (2,4), (4,2)} = 3;
%e T(4,3) = #{(3,3)} = 1;
%e T(4,4) = #{(4,4)} = 1.
%e The triangle begins: row sums
%e . 1: 1 1
%e . 2: 3 1 4
%e . 3: 7 1 1 9
%e . 4: 11 3 1 1 16
%e . 5: 19 3 1 1 1 25
%e . 6: 23 7 3 1 1 1 36
%e . 7: 35 7 3 1 1 1 1 49
%e . 8: 43 11 3 3 1 1 1 1 64
%e . 9: 55 11 7 3 1 1 1 1 1 81
%e . 10: 63 19 7 3 3 1 1 1 1 1 100
%e . 11: 83 19 7 3 3 1 1 1 1 1 1 121
%e . 12: 91 23 11 7 3 3 1 1 1 1 1 1 144
%e . 13: 115 23 11 7 3 3 1 1 1 1 1 1 1 169
%e . 14: 127 35 11 7 3 3 3 1 1 1 1 1 1 1 196
%e . 15: 143 35 19 7 7 3 3 1 1 1 1 1 1 1 1 225
%e . 16: 159 43 19 11 7 3 3 3 1 1 1 1 1 1 1 1 256
%e . 17: 191 43 19 11 7 3 3 3 1 1 1 1 1 1 1 1 1 289
%e . 18: 203 55 23 11 7 7 3 3 3 1 1 1 1 1 1 1 1 1 324 .
%t T[n_, k_] := 2 Total[EulerPhi[Range[Quotient[n, k]]]] - 1;
%t Table[T[n, k], {n, 1, 18}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 20 2021 *)
%o (Haskell)
%o a242114 n k = a242114_tabl !! (n-1) !! (k-1)
%o a242114_row n = a242114_tabl !! (n-1)
%o a242114_tabl = map (map a018805) a010766_tabl
%K nonn,tabl
%O 1,2
%A _Reinhard Zumkeller_, May 04 2014
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