OFFSET
0,4
COMMENTS
The complement is counted by A367222.
EXAMPLE
3 cannot be written as a nonnegative linear combination of 2, 4, and 5, so {2,4,5} is counted under a(6).
The a(2) = 1 through a(6) = 15 subsets:
{2} {2} {2} {2} {2}
{3} {3} {3} {3}
{4} {4} {4}
{3,4} {5} {5}
{3,4} {6}
{3,5} {3,4}
{4,5} {3,5}
{2,4,5} {3,6}
{4,5}
{4,6}
{5,6}
{2,4,5}
{2,4,6}
{2,5,6}
{4,5,6}
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]], combs[Length[#], Union[#]]=={}&]], {n, 0, 10}]
PROG
(Python)
from itertools import combinations
from sympy.utilities.iterables import partitions
def A367223(n):
c, mlist = 0, []
for m in range(1, n+1):
t = set()
for p in partitions(m):
t.add(tuple(sorted(p.keys())))
mlist.append([set(d) for d in t])
for k in range(1, n+1):
for w in combinations(range(1, n+1), k):
ws = set(w)
for s in mlist[k-1]:
if s <= ws:
break
else:
c += 1
return c # Chai Wah Wu, Nov 16 2023
CROSSREFS
The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free
-------------------------------------------
Triangles:
A116861 counts positive linear combinations of strict partitions of k.
A364916 counts linear combinations of strict partitions of k.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 14 2023
EXTENSIONS
a(14)-a(33) from Chai Wah Wu, Nov 15 2023
STATUS
approved