A186975 Irregular triangle T(n,k), n>=1, 1<=k<=A186971(n), read by rows: T(n,k) is the number of subsets of {1, 2, ..., n} containing n and having <=k pairwise coprime elements.

1, 1, 2, 1, 3, 4, 1, 3, 4, 1, 5, 10, 12, 1, 3, 4, 1, 7, 18, 26, 28, 1, 5, 11, 15, 16, 1, 7, 19, 29, 32, 1, 5, 10, 12, 1, 11, 42, 84, 110, 116, 1, 5, 11, 15, 16, 1, 13, 58, 137, 209, 242, 248, 1, 7, 21, 37, 46, 48, 1, 9, 30, 55, 69, 72, 1, 9, 33, 69, 98, 110, 112

Offset

1,3

Comments

T(n,k) = T(n,k-1) for k>A186971(n). The triangle contains all values of T up to the last element of each row that is different from its predecessor.

Links

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

Formula

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

Example

T(5,3) = 10 because there are 10 subsets of {1,2,3,4,5} containing n and having <=3 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}.
Triangle T(n,k) begins:
  1;
  1, 2;
  1, 3, 4;
  1, 3, 4;
  1, 5, 10, 12;
  1, 3, 4;
  1, 7, 18, 26, 28;

Maple

with(numtheory):
s:= proc(m, r) option remember; mul(`if`(i<r, i, 1), i=factorset(m)) end:
a:= proc(n) option remember; `if`(n<4, n, pi(n)-nops(factorset(n))+2) end:
b:= proc(t, n, k) option remember; local c, d, h;
      if k=0 or k>n 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) +`if`(k=0, 0, T(n, k-1))
    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]+If[k == 0, 0, t[n, k-1]]; Table[Table[t[n, k], {k, 1, a[n]}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 19 2013, translated from Maple *)

Crossrefs

Columns k=1-9 give: A000012, A039649 for n>1, A186987, A186988, A186989, A186990, A186991, A186992, A186993.

Rightmost elements of rows give A186973.

Cf. A186971, A186972.

Sequence in context: A276237 A059663 A338367 * A027422 A135086 A210561

Adjacent sequences: A186972 A186973 A186974 * A186976 A186977 A186978

Keyword

nonn,look,tabf

Author

Alois P. Heinz, Mar 02 2011

Status

approved