A322394 Heinz numbers of integer partitions with edge-connectivity 1.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 195, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Author

Gus Wiseman, Dec 06 2018

Keywords

• nonn

Comments

The first nonprime term is 195, which is the Heinz number of (6,3,2).

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

An integer partition has edge-connectivity 1 if the prime factorizations of the parts form a connected hypergraph that can be disconnected (or made empty) by removing a single part.

Links

• Wikipedia, k-edge-connected graph

Crossrefs

Cf. A007718, A013922, A054921, A056239, A095983, A112798, A218970, A304716, A305078, A305079, A322336, A322391, A322393.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
edgeConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[primeMS/@#]]!=1&]];
Select[Range[100],edgeConn[primeMS[#]]==1&]