A186981 -id:A186981 - cp's OEIS frontend

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

Alois P. Heinz, Mar 01 2011

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

			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;

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

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.

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);

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 *)

A186986 Number of 10-element subsets of {1, 2, ..., n} having pairwise coprime elements.

Original entry on oeis.org

8, 8, 16, 16, 24, 24, 268, 268, 1569, 1857, 1952, 2010, 2156, 2176, 9265, 9596, 10167, 10339, 35489, 35589, 103267, 107303, 111153, 115675, 297827, 301611, 442057, 451649, 484539, 503173, 1150127, 1164141, 1290703, 1320491, 1413885, 1471627

Offset: 23

Alois P. Heinz, Mar 02 2011

nonn

			a(24) = 8 because there are 8 10-element subsets of {1, 2, ..., 24} having pairwise coprime elements: {1,2,3,5,7,11,13,17,19,23}, {1,2,5,7,9,11,13,17,19,23}, {1,3,4,5,7,11,13,17,19,23}, {1,3,5,7,8,11,13,17,19,23}, {1,3,5,7,11,13,16,17,19,23}, {1,4,5,7,9,11,13,17,19,23}, {1,5,7,8,9,11,13,17,19,23}, {1,5,7,9,11,13,16,17,19,23}.

Alois P. Heinz, Table of n, a(n) for n = 23..200

Column 10 of triangle A186974. Partial sums of A186981.

cp's OEIS Frontend

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.

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