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A326083 Number of subsets of {1..n} containing all of their pairwise sums <= n. 92
1, 2, 3, 5, 7, 12, 16, 27, 37, 58, 80, 131, 171, 277, 380, 580, 785, 1250, 1655, 2616, 3516, 5344, 7257, 11353, 14931, 23204, 31379, 47511, 63778, 98681, 130503, 201357, 270038, 407429, 548090, 840171, 1110429, 1701872, 2284325, 3440337, 4601656 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The summands are allowed to be equal. The case where they must be distinct is A326080. If A007865 counts sum-free sets, this sequence counts sum-closed sets. This is different from sum-full sets (A093971).
From Gus Wiseman, Jul 08 2019: (Start)
Also the number of subsets of {1..n} containing no sum of any multiset of the elements. For example, the a(0) = 1 through a(6) = 16 subsets are:
{} {} {} {} {} {} {}
{1} {1} {1} {1} {1} {1}
{2} {2} {2} {2} {2}
{3} {3} {3} {3}
{2,3} {4} {4} {4}
{2,3} {5} {5}
{3,4} {2,3} {6}
{2,5} {2,3}
{3,4} {2,5}
{3,5} {3,4}
{4,5} {3,5}
{3,4,5} {4,5}
{4,6}
{5,6}
{3,4,5}
{4,5,6}
(End)
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100
FORMULA
For n > 0, a(n) = A103580(n) + 1.
EXAMPLE
The a(0) = 1 through a(6) = 16 subsets:
{} {} {} {} {} {} {}
{1} {2} {2} {3} {3} {4}
{1,2} {3} {4} {4} {5}
{2,3} {2,4} {5} {6}
{1,2,3} {3,4} {2,4} {3,6}
{2,3,4} {3,4} {4,5}
{1,2,3,4} {3,5} {4,6}
{4,5} {5,6}
{2,4,5} {2,4,6}
{3,4,5} {3,4,6}
{2,3,4,5} {3,5,6}
{1,2,3,4,5} {4,5,6}
{2,4,5,6}
{3,4,5,6}
{2,3,4,5,6}
{1,2,3,4,5,6}
The a(7) = 27 subsets:
{} {4} {36} {246} {2467} {24567} {234567} {1234567}
{5} {45} {356} {3467} {34567}
{6} {46} {367} {3567}
{7} {47} {456} {4567}
{56} {457}
{57} {467}
{67} {567}
MATHEMATICA
Table[Length[Select[Subsets[Range[n]], SubsetQ[#, Select[Plus@@@Tuples[#, 2], #<=n&]]&]], {n, 0, 10}]
CROSSREFS
Sequence in context: A319635 A179822 A319769 * A027959 A060730 A308928
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 05 2019
STATUS
approved

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Last modified April 21 12:56 EDT 2024. Contains 371870 sequences. (Running on oeis4.)