A051014 Number of nondividing sets on {1,2,...,n}.

1, 2, 3, 5, 7, 11, 14, 21, 27, 38, 52, 73, 90, 123, 159, 211, 263, 344, 413, 535, 658, 832, 1026, 1276, 1499, 1846, 2226, 2708, 3229, 3912, 4592, 5541, 6495, 7795, 9207, 10908, 12547, 14852, 17358, 20493, 23709, 27744, 31921, 37250, 43013, 49936, 57319, 66318

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn
• nice

Comments

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

Examples

			a(5) = 11 because there are 11 nondividing subsets of {1,2,3,4,5}: {}, {1}, {2}, {3}, {4}, {5}, {2,3}, {2,5}, {3,4}, {3,5}, {4,5}.
a(7) = 21: {}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {2,3}, {2,5}, {2,7}, {3,4}, {3,5}, {3,7}, {4,5}, {4,6}, {4,7}, {5,6}, {5,7}, {6,7}, {4,6,7}.
		

Links

• Eric Weisstein's World of Mathematics, Nondividing Set

Crossrefs

Row sums of A187489. Cf. A068063.

Programs

Maple

sums:= proc(s) option remember; local i, m;
          m:= max(s[]);
         `if`(m<1, {}, {m, seq([i,i+m][], i=sums(s minus {m}))})
       end:
b:= proc(i,s) option remember; local j, ok, t, si;
      if i<2 then 1
    else si:= s union {i};
         ok:= true;
         for j in sums(si) while ok do
           for t in si while ok do
             if irem(j, t)=0 and t<>j then ok:= false fi
           od
         od;
         b(i-1,s) +`if`(ok, b(i-1, si), 0)
      fi
    end:
a:= n-> `if`(n=0, 1, 1+b(n, {})):
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 08 2011

Mathematica

sums[s_] := sums[s] = Module[{m=Max[s]},
If[m<1, {},
  Join[{m},
  Sequence@@Table[{i,i+m}, {i,sums[DeleteCases[s,m]]}]]]
];
b[i_,s_] := b[i,s] = Module[{ ok,si,sij,sik},
If[ i<2, 1, si = Union[s,{i}];
ok = True;
For[j=1, j <= Length[sums[si]] && ok, j++,
  sij = sums[si][[j]];
  For[k=1, k <= Length[si] && ok, k++,
    If[Divisible[sij,sik=si[[k]]]&&sij!=sik,ok=False]]];
    b[i-1, s] + If[ok, b[i-1,si],0]
  ]
];
a[n_] := a[n] = If[n==0, 1, 1+b[n, {}]];
Table[ Print[ a[n] ]; a[n], {n,0,47}]
(* Jean-François Alcover, Oct 10 2012, after Alois P. Heinz *)

Extensions

More terms from David Wasserman, Feb 15 2002

a(41)-a(47) from Alois P. Heinz, Mar 08 2011