A186976 -id:A186976 - 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 *)

A186989 Number of subsets of {1, 2, ..., n} containing n and having <=5 pairwise coprime elements.

Original entry on oeis.org

1, 2, 4, 4, 12, 4, 28, 16, 32, 12, 110, 16, 209, 46, 69, 98, 507, 57, 828, 141, 277, 193, 1591, 163, 1289, 413, 1101, 441, 3785, 163, 5164, 1479, 1736, 1187, 2540, 609, 9561, 1879, 3086, 1304, 14298, 738, 18084, 3322, 3913, 3888, 25430

Offset: 1

Alois P. Heinz, Mar 03 2011

nonn

			a(10) = 12 because there are 12 subsets of {1, 2, ..., 10} containing 10 and having <=5 pairwise coprime elements: {10}, {1,10}, {3,10}, {7,10}, {9,10}, {1,3,10}, {1,7,10}, {1,9,10}, {3,7,10}, {7,9,10}, {1,3,7,10}, {1,7,9,10}. There is no subset with exactly 5 elements here.

Alois P. Heinz, Table of n, a(n) for n = 1..200

Column 5 of triangle A186975. Sum of A186988 and A186976.

A015698 Number of 5-tuples of different integers from [ 1,n ] with no common factors among pairs.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 3, 6, 6, 32, 33, 105, 114, 128, 157, 385, 400, 831, 886, 1001, 1076, 2004, 2074, 2817, 3022, 3655, 3880, 6379, 6449, 10020, 10978, 12063, 12791, 14417, 14767, 21746, 22977, 25038, 25860, 36660, 37110, 51088, 53449, 56199

Offset: 1

Olivier Gérard

nonn

Column 5 of triangle A186974. Partial sums of A186976.

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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A186989 Number of subsets of {1, 2, ..., n} containing n and having <=5 pairwise coprime elements.

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A015698 Number of 5-tuples of different integers from [ 1,n ] with no common factors among pairs.

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