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The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Given a finite multiset S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. This sequence lists all Heinz numbers of multisets S such that G(S) is a connected graph.

First differs from |A305052(n)| at a(169) = 1, A305052(169) = 0.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Given a finite multiset S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. If S is the integer partition with Heinz number n, a(n) is the number of connected components of G(S).

Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(LCM(S)), where omega = A001221 and LCM is least common multiple. Then a z-tree is a finite connected set of pairwise indivisible positive integers greater than 1 with clutter density -1.

This is a generalization to multiset systems of the usual definition of hypertree (viz. connected hypergraph F such that two distinct hyperedges of F intersect in at most a common vertex and such that every cycle of F is contained in a hyperedge).

If n is squarefree with k prime factors, then a(n) = A030019(k).

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The z-density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221 is number of distinct prime factors.

First nonnegative entry after a(1) = 0 is a(169) = 0.

A blob is a connected antichain of finite sets that cannot be capped by a hypertree with more than one branch. The weight of a blob is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices (see A275307).

Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(LCM(S)), where omega = A001221 and LCM is least common multiple. A z-forest is a finite set of pairwise indivisible positive integers greater than 1 such that all connected components are z-trees, meaning they have clutter density -1.

This is a generalization to multiset systems of the usual definition of hyperforest (viz. hypergraph F such that two distinct hyperedges of F intersect in at most a common vertex and such that every cycle of F is contained in a hyperedge).

If n is squarefree with k prime factors, then a(n) = A134954(k).

Differs from A324837 at positions {1, 180, 210, ...}. For example, a(210) = 55, A324837(210) = 49.

A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

Elements of a set-system are sometimes called edges. In an antichain, no edge is a subset or superset of any other edge. In a 2-vertex-connected set-system, at least two vertices must be removed to make the set-system disconnected. A blob is a connected, 2-vertex-connected antichain of finite, nonempty sets, or, equivalently, a 2-vertex-connected clutter.

Given a finite multiset S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. This sequence counts factorizations S such that G(S) is a connected graph.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Nov 07 2018

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

The density of a multiset partition is defined to be the sum of numbers of distinct elements in each part, minus the number of parts, minus the total number of distinct elements in the whole partition. A multiset partition is a tree if it has more than one part, is connected, and has density -1. A cap is a certain kind of non-transitive coarsening of a multiset partition. For example, the four caps of {{1,1},{1,2},{2,2}} are {{1,1},{1,2},{2,2}}, {{1,1},{1,2,2}}, {{1,1,2},{2,2}}, {{1,1,2,2}}. - Gus Wiseman, Feb 05 2021