A187262 Irregular triangle T(n,k), n>=1, 1<=k<=A036234(n), read by rows: T(n,k) is the number of nonempty subsets of {1, 2, ..., n} having <=k pairwise coprime elements.

1, 2, 3, 3, 6, 7, 4, 9, 11, 5, 14, 21, 23, 6, 17, 25, 27, 7, 24, 43, 53, 55, 8, 29, 54, 68, 71, 9, 36, 73, 97, 103, 10, 41, 83, 109, 115, 11, 52, 125, 193, 225, 231, 12, 57, 136, 208, 241, 247, 13, 70, 194, 345, 450, 489, 495, 14, 77, 215, 382, 496, 537, 543

Offset: 1

Author

Alois P. Heinz, Mar 07 2011

Keywords

• nonn
• tabf

Comments

T(n,k) = T(n,k-1) for k>A036234(n). The triangle contains all values of T up to the last element of each row that is different from its predecessor.

Examples

			T(5,3) = 21 because there are 21 nonempty subsets of {1,2,3,4,5} having <=3 pairwise coprime elements: {1}, {2}, {3}, {4}, {5}, {1,2}, {1,3}, {1,4}, {1,5}, {2,3}, {2,5}, {3,4}, {3,5}, {4,5}, {1,2,3}, {1,2,5}, {1,3,4}, {1,3,5}, {1,4,5}, {2,3,5}, {3,4,5}.
Irregular Triangle T(n,k) begins:
  1;
  2,  3;
  3,  6,  7;
  4,  9, 11;
  5, 14, 21, 23;
  6, 17, 25, 27;
  7, 24, 43, 53, 55;
		

Links

• Alois P. Heinz, Rows n = 1..200, flattened

Crossrefs

Columns k=1-10 give: A000027, A187263, A187264, A187265, A187266, A187267, A187268, A187269, A187270, A187271.

Rightmost elements of rows give A187106.

Cf. A036234, A186972, A186974, A186975.

Formula

T(n,k) = Sum_{i=1..n,j=1..k} A186972(i,j).

T(n,k) = Sum_{j=1..k} A186974(n,j).

T(n,k) = Sum_{i=1..n} A186975(i,k).