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}

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A way of writing n as a (presumed nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

First differs from A316402 at a(16) = 11 due to (7,5,3,1).

In sumset notation, number of subsets A of {1,...,n} such that the intersection of A and 2A is nonempty.

A variation of binary sum-full sets where parts can be re-used, this sequence counts subsets of {1..n} containing a part equal to the sum of two other (possibly equal) parts. The complement is counted by A007865. The non-binary version is A364914. For non-re-usable parts we have A088809. - Gus Wiseman, Aug 14 2023

These are partitions containing the sum of some 2-element submultiset of the parts, a variation of binary sum-full partitions where parts cannot be re-used, ranked by A364462. The complement is counted by A236912. The non-binary version is A237668. For re-usable parts we have A363225. - Gus Wiseman, Aug 10 2023

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

First differs from A151897 at a(7) = 61, A151897(7) = 60. The one subset counted under a(7) but not under A151897(7) is {1,2,4,7}. - Gus Wiseman, Jun 07 2019

These are partitions containing the sum of no 2-element submultiset of the parts, a variation of binary sum-free partitions where parts cannot be re-used, ranked by A364461. The complement is counted by A237113. The non-binary version is A237667. For re-usable parts we have A364345. - Gus Wiseman, Aug 09 2023

From Gus Wiseman, Aug 09 2023: (Start)

Includes all knapsack partitions (A108917), but first differs at a(12) = 28, A108917(12) = 25. The difference is accounted for by the non-knapsack partitions: (4332), (5331), (33222).

These are partitions not containing the sum of any non-singleton submultiset of the parts, a variation of non-binary sum-free partitions where parts cannot be re-used, ranked by A364531. The complement is counted by A237668. The binary version is A236912. For re-usable parts we have A364350.

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