A196721 Number of subsets of {1..n} (including empty set) such that the pairwise LCMs of elements are all distinct.

1, 2, 4, 8, 14, 28, 42, 84, 132, 236, 352, 704, 920, 1840, 2736, 3816, 5700, 11400, 15384, 30768, 39552, 54656, 81672, 163344, 196176, 362656, 542304, 930352, 1195168, 2390336, 2914304, 5828608, 8513920, 11674848, 17490432, 23484224, 28058816, 56117632, 84100800

Offset: 0

Author

Alois P. Heinz, Oct 05 2011

Keywords

• nonn

Examples

			a(4) = 14: {}, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,3,4}, {2,3,4}.
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A143823, A196719, A196720, 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(
       ilcm(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..10);

Mathematica

b[n_, s_] := b[n, s] = Module[{sn, m}, m = Length[s]; sn = Append[s, n]; If[n < 1, 1, b[n - 1, s] + If[m*(m + 1)/2 == Length @ Union @ Flatten @ Table[LCM [sn[[i]], sn[[j]]], {i, 1, m}, {j, i+1, m+1}], b[n-1, sn], 0]]]; a[n_] := a[n] = b[n-1, {n}] + If[n == 0, 0, a[n-1]]; Table[ Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 02 2017, translated from Maple *)

Extensions

Terms a(31) and beyond from Fausto A. C. Cariboni, Oct 18 2020