A196719 Number of subsets of {1..n} (including empty set) such that the pairwise GCDs of elements are all distinct.

1, 2, 4, 7, 11, 16, 24, 31, 40, 52, 68, 79, 102, 115, 140, 175, 201, 218, 265, 284, 336, 396, 446, 469, 547, 599, 662, 742, 837, 866, 1034, 1065, 1153, 1275, 1370, 1511, 1719, 1756, 1869, 2030, 2244, 2285, 2613, 2656, 2865, 3236, 3394, 3441, 3780, 3921, 4232

Offset: 0

Author

Alois P. Heinz, Oct 05 2011

Keywords

• nonn

Examples

			a(6) = 24: {}, {1}, {2}, {3}, {4}, {5}, {6}, {1,2}, {1,3}, {1,4}, {1,5}, {1,6}, {2,3}, {2,4}, {2,5}, {2,6}, {3,4}, {3,5}, {3,6}, {4,5}, {4,6}, {5,6}, {2,3,6}, {3,4,6}.
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1500 (terms 1..200 from Alois P. Heinz)

Crossrefs

Cf. A143823, A196720, A196721, A196722, A196723, A196724.

Programs

Maple

b:= proc(n, s) local sn, m;
      m:= nops(s);
      sn:= [s[], n];
      `if`(n<1, 1, b(n-1, s) +`if`(m*(m+1)/2 = nops(({seq(seq(
       igcd(sn[i], sn[j]), j=i+1..m+1), i=1..m)})), b(n-1, sn), 0))
    end:
a:= proc(n) option remember;
      b(n-1, [n]) +`if`(n=0, 0, a(n-1))
    end:
seq(a(n), n=0..50);

Mathematica

b[n_, s_] := b[n, s] = With[{m = Length[s], sn = Append[s, n]}, If[n<1, 1, b[n-1, s] + If[m*(m+1)/2 == Length[ Union @ Flatten @ Table[ Table[ GCD[ sn[[i]], sn[[j]]], {j, i+1, m+1}], {i, 1, m}]], b[n-1, sn], 0]]];
a[n_] := a[n] = b[n-1, {n}] + If[n == 0, 0, a[n-1]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 06 2017, translated from Maple *)