0,2

The number of subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different product is A325860(n). - Gus Wiseman, Jun 03 2019

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

a(6) = 58: from the 2^6=64 subsets of {1,2,3,4,5,6} only 6 do not have all the pairwise products of elements distinct: {1,2,3,6}, {2,3,4,6}, {1,2,3,4,6}, {1,2,3,5,6}, {2,3,4,5,6}, {1,2,3,4,5,6}.

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(

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..20);

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[ 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[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 31 2017, translated from Maple *)

Table[Length[Select[Subsets[Range[n]], UnsameQ@@Times@@@Subsets[#, {2}]&]], {n, 0, 10}] (* Gus Wiseman, Jun 03 2019 *)

Cf. A143823, A196719, A196720, A196721, A196722, A196723.

The subset case is A196724 (this sequence).

The maximal case is A325859.

The integer partition case is A325856.

The strict integer partition case is A325855.

Heinz numbers of the counterexamples are given by A325993.

Cf. A292886, A293627, A325860, A325861.

Sequence in context: A115909 A254940 A374495 * A056644 A007813 A289657

Adjacent sequences: A196721 A196722 A196723 * A196725 A196726 A196727

Alois P. Heinz, Oct 06 2011

Name edited by Gus Wiseman, Jun 03 2019

a(33)-a(35) from Fausto A. C. Cariboni, Oct 05 2020

approved