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Sets of this type may be called "positive combination-full".

Also subsets of {1..n} such that some element can be written as a (strictly) positive linear combination of the others.

Also subsets of {1..n} containing n whose greatest element cannot be written as a positive linear combination of the others.

Sets of this type may be called "positive combination-free".

Also subsets of {1..n} such that no element can be written as a (strictly) positive linear combination of the others.

From Gus Wiseman, Jan 22 2022: (Start)

Also the number of subsets of {1..n} containing n but without adjacent elements of quotient 1/2. The Heinz numbers of these sets are a subset of the squarefree terms of A320340. For example, the a(1) = 1 through a(6) = 19 subsets are:

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

{1,3} {1,4} {1,5} {1,6}

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

{1,3,4} {3,5} {4,6}

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

{1,3,5} {1,4,6}

{1,4,5} {1,5,6}

{2,3,5} {2,5,6}

{3,4,5} {3,4,6}

{1,3,4,5} {3,5,6}

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

{1,3,4,6}

{1,3,5,6}

{1,4,5,6}

{2,3,4,6}

{2,3,5,6}

{3,4,5,6}

{1,3,4,5,6}

{2,3,4,5,6}

(End)

Sets of this type may be called "positive combination-full".

Also subsets of {1..n} containing n whose greatest element can be written as a positive linear combination of the others.

A set is double-free if no element is twice any other element.

Also, the number of distinct numerical semigroups that are generated by some subset of {1, 2, ..., n} and have a finite complement in the positive integers.

This sequence can be calculated by a recursive algorithm:

Let B1 be an array of finite length, the "1" denotes that it is the first generation. Let B1' be the reversed version of B1. Let C be the element-wise product C = B1 * B1'. Then B2 is a concatenation of taking each element of B1 and add all divisors of the corresponding element in C. If we start with B1 = {1} then we get this sequence of arrays: B2 = {2}, B3 = {3, 4, 6}, ... . a(n) is the length of the array Bn. In short the length of Bn+1 and so a(n+1) is the sum over A000005(Bn * Bn').

The transform used in the definition of this sequence is its own inverse, so if c = S(b) then b = S(c). The eigensequence is 2^n = S(2^n).

There exist some transformation pairs of infinite sequences in the database:

A000027 <--> A000142; A000079 <--> A000079; A000045 <--> A001654;

A026549 <--> A038754; A100071 <--> A001405; A058295 <--> A------;

A111286 <--> A098011; A093968 <--> A205825; A166447 <--> A------;

A098558 <--> A004277; A010551 <--> A087811; A100538 <--> A329227;

A079352 <--> A------; A082458 <--> A------; A008233 <--> A264635;

A138278 <--> A------; A006501 <--> A264557; A336496 <--> A------;

A019464 <--> A------; A062112 <--> A------; A171647 <--> A359039;

A279312 <--> A------; A031923 <--> A------.

These transformation pairs are conjectured:

A137326 <--> A------; A066332 <--> A300902; A208147 <--> A308546;

A057895 <--> A------; A349080 <--> A------; A019442 <--> A------;

A349079 <--> A------.

("A------" means not yet in the database.)

Some sequences in the lists above may need offset adjustment to force a beginning with 1,2,... in the transformation.

If we allowed signed rational numbers, further interesting transformation pairs could be observed. For example, 1/n will transform into factorials with alternating sign. 2^(-n) transforms into ones with alternating sign and 1/A000045(n) into A000045 with alternating sign.