cp's OEIS Frontend

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

%I A196721 #17 Oct 18 2020 05:50:36
%S A196721 1,2,4,8,14,28,42,84,132,236,352,704,920,1840,2736,3816,5700,11400,
%T A196721 15384,30768,39552,54656,81672,163344,196176,362656,542304,930352,
%U A196721 1195168,2390336,2914304,5828608,8513920,11674848,17490432,23484224,28058816,56117632,84100800
%N A196721 Number of subsets of {1..n} (including empty set) such that the pairwise LCMs of elements are all distinct.
%H A196721 Fausto A. C. Cariboni, <a href="/A196721/b196721.txt">Table of n, a(n) for n = 0..50</a>
%e A196721 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}.
%p A196721 b:= proc(n, s) local sn, m;
%p A196721       m:= nops(s);
%p A196721       sn:= [s[], n];
%p A196721       `if`(n<1, 1, b(n-1, s) +`if`(m*(m+1)/2 = nops(({seq(seq(
%p A196721        ilcm(sn[i], sn[j]), j=i+1..m+1), i=1..m)})), b(n-1, sn), 0))
%p A196721     end:
%p A196721 a:= proc(n) option remember;
%p A196721       b(n-1, [n]) +`if`(n=0, 0, a(n-1))
%p A196721     end:
%p A196721 seq(a(n), n=0..10);
%t A196721 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 *)
%Y A196721 Cf. A143823, A196719, A196720, A196722, A196723, A196724.
%K A196721 nonn
%O A196721 0,2
%A A196721 _Alois P. Heinz_, Oct 05 2011
%E A196721 Terms a(31) and beyond from _Fausto A. C. Cariboni_, Oct 18 2020