A185348 -id:A185348 - 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 *)

A186988 Number of subsets of {1, 2, ..., n} containing n and having <=4 pairwise coprime elements.

Original entry on oeis.org

1, 2, 4, 4, 12, 4, 26, 15, 29, 12, 84, 15, 137, 37, 55, 69, 279, 42, 397, 86, 162, 118, 663, 93, 546, 208, 468, 216, 1286, 93, 1593, 521, 651, 459, 914, 259, 2582, 648, 1025, 482, 3498, 288, 4106, 961, 1163, 1116, 5316, 641, 4326, 1033

Offset: 1

Alois P. Heinz, Mar 03 2011

nonn

			a(5) = 12 because there are 12 subsets of {1,2,3,4,5} containing 5 and having <=4 pairwise coprime elements: {5}, {1,5}, {2,5}, {3,5}, {4,5}, {1,2,5}, {1,3,5}, {1,4,5}, {2,3,5}, {3,4,5}, {1,2,3,5}, {1,3,4,5}.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Column 4 of triangle A186975. Sum of A186987 and A185348.

A015623 Quadruples of different integers from [ 1,n ] with no common factors between pairs.

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 10, 14, 24, 26, 68, 72, 151, 167, 192, 228, 411, 431, 708, 758, 858, 928, 1419, 1475, 1874, 2013, 2353, 2499, 3515, 3571, 4856, 5254, 5747, 6089, 6795, 6979, 9129, 9630, 10437, 10800, 13768, 13978, 17500, 18275, 19210, 20114, 24734

Offset: 1

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Partial sums of A185348, Column 4 of triangle A186974.

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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A186988 Number of subsets of {1, 2, ..., n} containing n and having <=4 pairwise coprime elements.

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A015623 Quadruples of different integers from [ 1,n ] with no common factors between pairs.

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