A186972 Irregular triangle T(n,k), n>=1, 1<=k<=A186971(n), read by rows: T(n,k) is the number of k-element subsets of {1, 2, ..., n} containing n and having pairwise coprime elements.

1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4, 5, 2, 1, 2, 1, 1, 6, 11, 8, 2, 1, 4, 6, 4, 1, 1, 6, 12, 10, 3, 1, 4, 5, 2, 1, 10, 31, 42, 26, 6, 1, 4, 6, 4, 1, 1, 12, 45, 79, 72, 33, 6, 1, 6, 14, 16, 9, 2, 1, 8, 21, 25, 14, 3, 1, 8, 24, 36, 29, 12, 2, 1, 16, 79, 183, 228, 157, 56, 8, 1, 6, 15, 20, 15, 6, 1

Offset: 1

Author

Alois P. Heinz, Mar 01 2011

Keywords

• nonn
• tabf
• look

Comments

T(n,k) = 0 for k>A186971(n). The triangle contains all positive values of T.

Examples

			T(5,3) = 5 because there are 5 3-element subsets of {1,2,3,4,5} containing 5 and having pairwise coprime elements: {1,2,5}, {1,3,5}, {1,4,5}, {2,3,5}, {3,4,5}.
Irregular Triangle T(n,k) begins:
  1;
  1, 1;
  1, 2,  1;
  1, 2,  1;
  1, 4,  5, 2;
  1, 2,  1;
  1, 6, 11, 8, 2;
		

Links

• Alois P. Heinz, Rows n = 1..220, flattened

Crossrefs

Columns k=1-10 give: A000012, A000010 (for n>1), A185953, A185348, A186976, A186977, A186978, A186979, A186980, A186981.

Rightmost elements of rows give A186994.

Row sums are A186973.

Cf. A186971.

Programs

Maple

with(numtheory):
s:= proc(m,r) option remember; mul(`if`(in then 0
    elif k=1 then 1
    elif k=2 and t=n then `if`(n<2, 0, phi(n))
    else c:= 0;
         d:= 2-irem(t,2);
         for h from 1 to n-1 by d do
           if igcd(t, h)=1 then c:= c +b(s(t*h, h), h, k-1) fi
         od; c
      fi
end:
T:= proc(n,k) option remember; b(s(n,n),n,k) end:
seq(seq(T(n, k), k=1..a(n)), n=1..20);

Mathematica

s[m_, r_] := s[m, r] = Product[If[i < r, i, 1], {i, FactorInteger[m][[All, 1]]}]; a[n_] := a[n] = If[n < 4, n, PrimePi[n] - Length[FactorInteger[n]]+2]; b[t_, n_, k_] := b[t, n, k] = Module[{c, d, h}, Which[k == 0 || k > n, 0, k == 1, 1, k == 2 && t == n, If[n < 2, 0, EulerPhi[n]], True, c = 0; d = 2-Mod[t, 2]; For[h = 1, h <= n-1, h = h+d, If[GCD[t, h] == 1, c = c+b[s[t*h, h], h, k-1]]]; c]]; t[n_, k_] := t[n, k] = b[s[n, n], n, k]; Table[Table[t[n, k], {k, 1, a[n]}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)