A322387 - OEIS

%I #10 Jun 20 2020 09:00:39

%S 0,0,0,0,0,1,0,1,1,3,1,6,2,10,8,13,9,26,14,35,28,50,37,77,54,101,84,

%T 138,110,205,149,252,222,335,287,455,375,577,522,740,657,985

%N Number of 2-vertex-connected integer partitions of n.

%C An integer partition is 2-vertex-connected if the prime factorizations of the parts form a connected hypergraph that is still connected if any single prime number is divided out of all the parts (and any parts then equal to 1 are removed).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/K-vertex-connected_graph">k-vertex-connected graph</a>

%e The a(14) = 10 2-vertex-connected integer partitions:

%e   (14)  (8,6)   (6,4,4)   (6,3,3,2)  (6,2,2,2,2)

%e         (10,4)  (6,6,2)   (6,4,2,2)

%e         (12,2)  (10,2,2)

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t 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]]]]]]]]];

%t vertConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@primeMS/@y],Function[del,Length[csm[DeleteCases[DeleteCases[primeMS/@y,Alternatives@@del,{2}],{}]]]!=1]]];

%t Table[Length[Select[IntegerPartitions[n],vertConn[#]>1&]],{n,30}]

%Y Cf. A013922, A095983, A218970, A275307, A304714, A304716, A305078, A305079, A322335, A322336, A322337, A322338, A322388, A322389, A322390.

%K nonn,more

%O 1,10

%A _Gus Wiseman_, Dec 05 2018

%E a(41)-a(42) from _Jinyuan Wang_, Jun 20 2020