OFFSET
1,10
COMMENTS
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).
LINKS
EXAMPLE
The a(14) = 10 2-vertex-connected integer partitions:
(14) (8,6) (6,4,4) (6,3,3,2) (6,2,2,2,2)
(10,4) (6,6,2) (6,4,2,2)
(12,2) (10,2,2)
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]]]]]]]]];
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]]];
Table[Length[Select[IntegerPartitions[n], vertConn[#]>1&]], {n, 30}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 05 2018
EXTENSIONS
a(41)-a(42) from Jinyuan Wang, Jun 20 2020
STATUS
approved