A242114 Triangle read by rows: T(n,k) = number of pairs (x,y) in {1..n}X{1..n} with gcd(x,y) = k.

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, 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, 1, 1, 1, 1, 91, 23, 11, 7, 3, 3, 1, 1, 1, 1, 1, 1, 115, 23

Offset: 1

Author

Reinhard Zumkeller, May 04 2014

Keywords

• nonn
• tabl

Comments

T(n,1) = A018805(n);

sum(T(n,k): k = 1..n) = A000290(n);

sum(T(n,k): k = 2..n) = A100613(n);

T(floor(n/k),1) = A018805(n).

Examples

			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;
T(4,2) = #{(2,2), (2,4), (4,2)} = 3;
T(4,3) = #{(3,3)} = 1;
T(4,4) = #{(4,4)} = 1.
The triangle begins:                                            row sums
.   1:    1                                                            1
.   2:    3   1                                                        4
.   3:    7   1   1                                                    9
.   4:   11   3   1   1                                               16
.   5:   19   3   1   1  1                                            25
.   6:   23   7   3   1  1  1                                         36
.   7:   35   7   3   1  1  1  1                                      49
.   8:   43  11   3   3  1  1  1  1                                   64
.   9:   55  11   7   3  1  1  1  1  1                                81
.  10:   63  19   7   3  3  1  1  1  1  1                            100
.  11:   83  19   7   3  3  1  1  1  1  1  1                         121
.  12:   91  23  11   7  3  3  1  1  1  1  1  1                      144
.  13:  115  23  11   7  3  3  1  1  1  1  1  1  1                   169
.  14:  127  35  11   7  3  3  3  1  1  1  1  1  1  1                196
.  15:  143  35  19   7  7  3  3  1  1  1  1  1  1  1  1             225
.  16:  159  43  19  11  7  3  3  3  1  1  1  1  1  1  1  1          256
.  17:  191  43  19  11  7  3  3  3  1  1  1  1  1  1  1  1  1       289
.  18:  203  55  23  11  7  7  3  3  3  1  1  1  1  1  1  1  1  1    324 .
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of table, flattened

Programs

Haskell

a242114 n k = a242114_tabl !! (n-1) !! (k-1)
a242114_row n = a242114_tabl !! (n-1)
a242114_tabl = map (map a018805) a010766_tabl

Mathematica

T[n_, k_] := 2 Total[EulerPhi[Range[Quotient[n, k]]]] - 1;
Table[T[n, k], {n, 1, 18}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 20 2021 *)

Formula

T(n,k) = A018805(A010766(n,k));