cp's OEIS Frontend

More precisely, subsets of {1,...,n} containing no solutions of x+y=z.

There are two proofs that a(n) is 2^{n/2}(1+o(1)), as Paul Erdős and I conjectured.

In sumset notation, number of subsets A of {1,...,n} such that the intersection of A and 2A is empty. Using the Mathematica program, all such subsets can be printed. - T. D. Noe, Apr 20 2004

The Sapozhenko paper has many additional references.

If this sequence counts sum-free sets, then A326083 counts sum-closed sets, which is different from sum-full sets (A093971). - Gus Wiseman, Jul 08 2019

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)

a(n) = 2^n - A085489(n); a non-sum-free subset contains at least one subset {u,v, w} with w=u+v.

A variation of binary sum-full sets where parts cannot be re-used, this sequence counts subsets of {1..n} with an element equal to the sum of two distinct others. The complement is counted by A085489. The non-binary version is A364534. For re-usable parts we have A093971. - Gus Wiseman, Aug 10 2023

A variation of non-binary combination-full sets where parts can be re-used. The complement is counted by A326083. The binary version is A093971. For non-re-usable parts we have A364534. First differences are A365046.

Equivalently, a(n) is the number of subsets of {1..n} containing the sum of any two distinct elements whose sum is <= n.

The summands must be distinct. The case where they are allowed to be equal is A326083.

If A151897 counts sum-free sets, this sequence counts sum-closed sets. This is different from sum-full sets (A093971).

From Gus Wiseman, Aug 28 2023: (Start)

Appears to be the number of subsets of {1..n} containing n such that no element can be written as a nonnegative linear combination of the others, first differences of A326083. For example, the a(1) = 1 through a(8) = 10 subsets are:

{1} {2} {3} {4} {5} {6} {7} {8}

{2,3} {3,4} {2,5} {4,6} {2,7} {3,8}

{3,5} {5,6} {3,7} {5,8}

{4,5} {4,5,6} {4,7} {6,8}

{3,4,5} {5,7} {7,8}

{6,7} {3,7,8}

{3,5,7} {5,6,8}

{4,5,7} {5,7,8}

{4,6,7} {6,7,8}

{5,6,7} {5,6,7,8}

{4,5,6,7}

Note that these subsets do not all generate numerical semigroups, as their GCD is unrestricted, cf. A358392. The complement is counted by A365046, first differences of A364914.

(End)

Number of different values of A007947(m) when A056239(m) is equal to n.

From Gus Wiseman, Sep 11 2023: (Start)

Also the number of finite sets of positive integers that can be linearly combined using all positive coefficients to obtain n. For example, the a(1) = 1 through a(7) = 12 sets are:

{1} {1} {1} {1} {1} {1} {1}

{2} {3} {2} {5} {2} {7}

{1,2} {4} {1,2} {3} {1,2}

{1,2} {1,3} {6} {1,3}

{1,3} {1,4} {1,2} {1,4}

{2,3} {1,3} {1,5}

{1,4} {1,6}

{1,5} {2,3}

{2,4} {2,5}

{1,2,3} {3,4}

{1,2,3}

{1,2,4}

Cf. A365073, A365311, A365312, A365322, A365380.

(End)

Includes all subsets containing both 1 and n.