OFFSET
0,4
COMMENTS
Includes all subsets containing both 1 and n.
LINKS
S. R. Finch, Monoids of natural numbers, March 17, 2009.
FORMULA
a(n+1) = 2^n - A124506(n).
EXAMPLE
The subset {3,4,10} has 10 = 2*3 + 1*4 so is counted under a(10).
The a(0) = 0 through a(5) = 11 subsets:
. . {1,2} {1,3} {1,4} {1,5}
{1,2,3} {2,4} {1,2,5}
{1,2,4} {1,3,5}
{1,3,4} {1,4,5}
{2,3,4} {2,3,5}
{1,2,3,4} {2,4,5}
{1,2,3,5}
{1,2,4,5}
{1,3,4,5}
{2,3,4,5}
{1,2,3,4,5}
MATHEMATICA
combs[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n]], MemberQ[#, n]&&Or@@Table[combs[#[[k]], Union[Delete[#, k]]]!={}, {k, Length[#]}]&]], {n, 0, 10}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 24 2023
STATUS
approved